https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Hung&feedformat=atomUW-Math Wiki - User contributions [en]2019-12-16T13:29:53ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=18490Colloquia2019-11-25T17:09:00Z<p>Hung: /* Mathematics Colloquium */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6 '''Room 911'''<br />
| Will Sawin (Columbia)<br />
| [[#Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] ]]<br />
| Marshall<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| [http://mate.dm.uba.ar/~alidick/ Alicia Dickenstein] (Buenos Aires)<br />
|[[#Alicia Dickenstein (Buenos Aires)| Algebra and geometry in the study of enzymatic cascades ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| [https://math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke)<br />
|[[#Jianfeng Lu (Duke) | How to "localize" the computation?]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 26 '''Thursday 3-4 pm Room 911'''<br />
| [http://eugeniacheng.com/ Eugenia Cheng] (School of the Art Institute of Chicago)<br />
| [[#Eugenia Cheng (School of the Art Institute of Chicago)| Character vs gender in mathematics and beyond ]]<br />
| Marshall / Friends of UW Madison Libraries<br />
|<br />
|-<br />
|Sept 27<br />
|<br />
|<br />
|-<br />
|Oct 4<br />
|<br />
|<br />
|-<br />
|Oct 11<br />
| Omer Mermelstein (Madison)<br />
| [[#Omer Mermelstein (Madison)| Generic flat pregeometries ]]<br />
|Andrews<br />
|<br />
|-<br />
|Oct 18<br />
| Shamgar Gurevich (Madison)<br />
| [[#Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields ]]<br />
| Marshall<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Elchanan Mossel (MIT)<br />
|Distinguished Lecture<br />
|Roch<br />
|-<br />
|Nov 8<br />
|Jose Rodriguez (UW-Madison)<br />
|[[#Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees]]<br />
|Erman<br />
|-<br />
|Nov 13 '''Wednesday 4-5pm'''<br />
|Ananth Shankar (MIT)<br />
|Exceptional splitting of abelian surfaces<br />
|-<br />
|Nov 20 '''Wednesday 4-5pm'''<br />
|Franca Hoffman (Caltech)<br />
|[[#Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis]]<br />
|Smith<br />
|-<br />
|Nov 22<br />
| Jeffrey Danciger (UT Austin)<br />
| [[#Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture"]]<br />
| Kent<br />
|-<br />
|Nov 25 '''Monday 4-5 pm Room 911'''<br />
|Tatyana Shcherbina (Princeton)<br />
| [[# Tatyana Shcherbina (Princeton)| "Random matrix theory and supersymmetry techniques"]]<br />
|Roch<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 2 '''Monday 4-5pm'''<br />
|Tingran Gao (University of Chicago)<br />
| [[#Tingran Gao (University of Chicago)| "Manifold Learning on Fibre Bundles"]]<br />
|Smith<br />
|-<br />
|Dec 4 '''Wednesday 4-5 pm'''<br />
|Andrew Zimmer<br />
| Intrinsic and extrinsic geometries in several complex variables<br />
|Gong<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 9 '''Monday 4-5 pm'''<br />
|Hui Yu (Columbia)<br />
| [[#Hui Yu (Columbia)| Singular sets in obstacle problems]]<br />
|Tran<br />
|-<br />
|Dec 11 '''Wednesday'''<br />
|Nick Higham (Manchester)<br />
|LAA lecture<br />
|Brualdi<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Jan 31<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 7<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 14<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 21<br />
|Shai Evra (IAS)<br />
|<br />
|Gurevich<br />
|<br />
|-<br />
|Feb 28<br />
|Brett Wick (Washington University, St. Louis)<br />
|<br />
|Seeger<br />
|-<br />
|March 6<br />
| Jessica Fintzen (Michigan)<br />
|<br />
|Marshall<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|(Moduli Spaces Conference)<br />
|<br />
|Boggess, Sankar<br />
|-<br />
|April 3<br />
|Caroline Turnage-Butterbaugh (Carleton College)<br />
|<br />
|Marshall<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|Song Sun (Berkeley)<br />
|<br />
|Huang<br />
|-<br />
|April 24<br />
|Natasa Sesum (Rutgers University)<br />
|<br />
|Angenent<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Will Sawin (Columbia)===<br />
<br />
Title: On Chowla's Conjecture over F_q[T]<br />
<br />
Abstract: The Mobius function in number theory is a sequences of 1s, <br />
-1s, and 0s, which is simple to define and closely related to the <br />
prime numbers. Its behavior seems highly random. Chowla's conjecture <br />
is one precise formalization of this randomness, and has seen recent <br />
work by Matomaki, Radziwill, Tao, and Teravainen making progress on <br />
it. In joint work with Mark Shusterman, we modify this conjecture by <br />
replacing the natural numbers parameterizing this sequence with <br />
polynomials over a finite field. Under mild conditions on the finite <br />
field, we are able to prove a strong form of this conjecture. The <br />
proof is based on taking a geometric perspective on the problem, and <br />
succeeds because we are able to simplify the geometry using a trick <br />
based on the strange properties of polynomial derivatives over finite <br />
fields.<br />
<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
<br />
===Alicia Dickenstein (Buenos Aires)===<br />
<br />
Title: Algebra and geometry in the study of enzymatic cascades<br />
<br />
Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible.<br />
I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.<br />
<br />
<br />
=== Jianfeng Lu (Duke) ===<br />
Title: How to ``localize" the computation?<br />
<br />
It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.<br />
<br />
===Eugenia Cheng (School of the Art Institute of Chicago)===<br />
<br />
Title: Character vs gender in mathematics and beyond<br />
<br />
Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.<br />
<br />
===Omer Mermelstein (Madison)===<br />
<br />
Title: Generic flat pregeometries<br />
<br />
Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure.<br />
The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.<br />
<br />
<br />
===Shamgar Gurevich (Madison)===<br />
<br />
Title: Harmonic Analysis on GL(n) over Finite Fields.<br />
<br />
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:<br />
<br />
trace(ρ(g)) / dim(ρ),<br />
<br />
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.<br />
<br />
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. <br />
<br />
This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.<br />
<br />
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)<br />
<br />
<br />
===Jose Rodriguez (UW-Madison)===<br />
Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science,<br />
engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient<br />
descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer.<br />
An alternative method is to determine the critical points of an objective function on the model.<br />
In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree).<br />
In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover,<br />
I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for<br />
determining critical points of the squared Euclidean distance function.<br />
<br />
<br />
===Ananth Shankar (MIT)===<br />
An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfaces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.<br />
<br />
===Franca Hoffman (Caltech)===<br />
<br />
Title: Gradient Flows: From PDE to Data Analysis.<br />
<br />
Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches.<br />
In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs.<br />
In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.<br />
<br />
=== Jeffrey Danciger (UT Austin) ===<br />
<br />
Title: Affine geometry and the Auslander Conjecture<br />
<br />
Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. <br />
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm&uuml;ller theory will enter the picture.<br />
<br />
=== Tatyana Shcherbina (Princeton) ===<br />
<br />
Title: Random matrix theory and supersymmetry techniques<br />
<br />
Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a special structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices<br />
whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. <br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
=== Tingran Gao (University of Chicago) ===<br />
<br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
Abstract: Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.<br />
<br />
=== Andrew Zimmer (LSU) ===<br />
<br />
Title: Intrinsic and extrinsic geometries in several complex variables<br />
<br />
Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.<br />
<br />
=== Hui Yu (Columbia) ===<br />
<br />
Title: Singular sets in obstacle problems<br />
<br />
Abstract: One of the most important free boundary problems is the obstacle problem. The regularity of its free boundary has been studied for over half a century. In this talk, we review some classical results as well as exciting new developments. In particular, we discuss the recent resolution of the regularity of the singular set for the fully nonlinear obstacle problem. This talk is based on a joint work with Ovidiu Savin at Columbia University.<br />
<br />
== Future Colloquia ==<br />
[[Colloquia/Fall 2020| Fall 2020]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=18489Colloquia2019-11-25T17:08:36Z<p>Hung: /* Mathematics Colloquium */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6 '''Room 911'''<br />
| Will Sawin (Columbia)<br />
| [[#Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] ]]<br />
| Marshall<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| [http://mate.dm.uba.ar/~alidick/ Alicia Dickenstein] (Buenos Aires)<br />
|[[#Alicia Dickenstein (Buenos Aires)| Algebra and geometry in the study of enzymatic cascades ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| [https://math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke)<br />
|[[#Jianfeng Lu (Duke) | How to "localize" the computation?]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 26 '''Thursday 3-4 pm Room 911'''<br />
| [http://eugeniacheng.com/ Eugenia Cheng] (School of the Art Institute of Chicago)<br />
| [[#Eugenia Cheng (School of the Art Institute of Chicago)| Character vs gender in mathematics and beyond ]]<br />
| Marshall / Friends of UW Madison Libraries<br />
|<br />
|-<br />
|Sept 27<br />
|<br />
|<br />
|-<br />
|Oct 4<br />
|<br />
|<br />
|-<br />
|Oct 11<br />
| Omer Mermelstein (Madison)<br />
| [[#Omer Mermelstein (Madison)| Generic flat pregeometries ]]<br />
|Andrews<br />
|<br />
|-<br />
|Oct 18<br />
| Shamgar Gurevich (Madison)<br />
| [[#Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields ]]<br />
| Marshall<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Elchanan Mossel (MIT)<br />
|Distinguished Lecture<br />
|Roch<br />
|-<br />
|Nov 8<br />
|Jose Rodriguez (UW-Madison)<br />
|[[#Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees]]<br />
|Erman<br />
|-<br />
|Nov 13 '''Wednesday 4-5pm'''<br />
|Ananth Shankar (MIT)<br />
|Exceptional splitting of abelian surfaces<br />
|-<br />
|Nov 20 '''Wednesday 4-5pm'''<br />
|Franca Hoffman (Caltech)<br />
|[[#Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis]]<br />
|Smith<br />
|-<br />
|Nov 22<br />
| Jeffrey Danciger (UT Austin)<br />
| [[#Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture"]]<br />
| Kent<br />
|-<br />
|Nov 25 '''Monday 4-5 pm Room 911'''<br />
|Tatyana Shcherbina (Princeton)<br />
| [[# Tatyana Shcherbina (Princeton)| "Random matrix theory and supersymmetry techniques"]]<br />
|Roch<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 2 '''Monday 4-5pm'''<br />
|Tingran Gao (University of Chicago)<br />
| [[#Tingran Gao (University of Chicago)| "Manifold Learning on Fibre Bundles"]]<br />
|Smith<br />
|-<br />
|Dec 4 '''Wednesday 4-5 pm'''<br />
|Andrew Zimmer<br />
| Intrinsic and extrinsic geometries in several complex variables<br />
|Gong<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 9 '''Monday 4-5 pm'''<br />
|Hui Yu (Columbia)<br />
| [[#Hui Yu (Columbia)| "Singular sets in obstacle problems"]]<br />
|Tran<br />
|-<br />
|Dec 11 '''Wednesday'''<br />
|Nick Higham (Manchester)<br />
|LAA lecture<br />
|Brualdi<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Jan 31<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 7<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 14<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 21<br />
|Shai Evra (IAS)<br />
|<br />
|Gurevich<br />
|<br />
|-<br />
|Feb 28<br />
|Brett Wick (Washington University, St. Louis)<br />
|<br />
|Seeger<br />
|-<br />
|March 6<br />
| Jessica Fintzen (Michigan)<br />
|<br />
|Marshall<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|(Moduli Spaces Conference)<br />
|<br />
|Boggess, Sankar<br />
|-<br />
|April 3<br />
|Caroline Turnage-Butterbaugh (Carleton College)<br />
|<br />
|Marshall<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|Song Sun (Berkeley)<br />
|<br />
|Huang<br />
|-<br />
|April 24<br />
|Natasa Sesum (Rutgers University)<br />
|<br />
|Angenent<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Will Sawin (Columbia)===<br />
<br />
Title: On Chowla's Conjecture over F_q[T]<br />
<br />
Abstract: The Mobius function in number theory is a sequences of 1s, <br />
-1s, and 0s, which is simple to define and closely related to the <br />
prime numbers. Its behavior seems highly random. Chowla's conjecture <br />
is one precise formalization of this randomness, and has seen recent <br />
work by Matomaki, Radziwill, Tao, and Teravainen making progress on <br />
it. In joint work with Mark Shusterman, we modify this conjecture by <br />
replacing the natural numbers parameterizing this sequence with <br />
polynomials over a finite field. Under mild conditions on the finite <br />
field, we are able to prove a strong form of this conjecture. The <br />
proof is based on taking a geometric perspective on the problem, and <br />
succeeds because we are able to simplify the geometry using a trick <br />
based on the strange properties of polynomial derivatives over finite <br />
fields.<br />
<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
<br />
===Alicia Dickenstein (Buenos Aires)===<br />
<br />
Title: Algebra and geometry in the study of enzymatic cascades<br />
<br />
Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible.<br />
I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.<br />
<br />
<br />
=== Jianfeng Lu (Duke) ===<br />
Title: How to ``localize" the computation?<br />
<br />
It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.<br />
<br />
===Eugenia Cheng (School of the Art Institute of Chicago)===<br />
<br />
Title: Character vs gender in mathematics and beyond<br />
<br />
Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.<br />
<br />
===Omer Mermelstein (Madison)===<br />
<br />
Title: Generic flat pregeometries<br />
<br />
Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure.<br />
The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.<br />
<br />
<br />
===Shamgar Gurevich (Madison)===<br />
<br />
Title: Harmonic Analysis on GL(n) over Finite Fields.<br />
<br />
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:<br />
<br />
trace(ρ(g)) / dim(ρ),<br />
<br />
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.<br />
<br />
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. <br />
<br />
This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.<br />
<br />
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)<br />
<br />
<br />
===Jose Rodriguez (UW-Madison)===<br />
Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science,<br />
engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient<br />
descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer.<br />
An alternative method is to determine the critical points of an objective function on the model.<br />
In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree).<br />
In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover,<br />
I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for<br />
determining critical points of the squared Euclidean distance function.<br />
<br />
<br />
===Ananth Shankar (MIT)===<br />
An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfaces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.<br />
<br />
===Franca Hoffman (Caltech)===<br />
<br />
Title: Gradient Flows: From PDE to Data Analysis.<br />
<br />
Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches.<br />
In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs.<br />
In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.<br />
<br />
=== Jeffrey Danciger (UT Austin) ===<br />
<br />
Title: Affine geometry and the Auslander Conjecture<br />
<br />
Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. <br />
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm&uuml;ller theory will enter the picture.<br />
<br />
=== Tatyana Shcherbina (Princeton) ===<br />
<br />
Title: Random matrix theory and supersymmetry techniques<br />
<br />
Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a special structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices<br />
whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. <br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
=== Tingran Gao (University of Chicago) ===<br />
<br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
Abstract: Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.<br />
<br />
=== Andrew Zimmer (LSU) ===<br />
<br />
Title: Intrinsic and extrinsic geometries in several complex variables<br />
<br />
Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.<br />
<br />
=== Hui Yu (Columbia) ===<br />
<br />
Title: Singular sets in obstacle problems<br />
<br />
Abstract: One of the most important free boundary problems is the obstacle problem. The regularity of its free boundary has been studied for over half a century. In this talk, we review some classical results as well as exciting new developments. In particular, we discuss the recent resolution of the regularity of the singular set for the fully nonlinear obstacle problem. This talk is based on a joint work with Ovidiu Savin at Columbia University.<br />
<br />
== Future Colloquia ==<br />
[[Colloquia/Fall 2020| Fall 2020]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=18488Colloquia2019-11-25T17:07:28Z<p>Hung: /* Mathematics Colloquium */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6 '''Room 911'''<br />
| Will Sawin (Columbia)<br />
| [[#Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] ]]<br />
| Marshall<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| [http://mate.dm.uba.ar/~alidick/ Alicia Dickenstein] (Buenos Aires)<br />
|[[#Alicia Dickenstein (Buenos Aires)| Algebra and geometry in the study of enzymatic cascades ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| [https://math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke)<br />
|[[#Jianfeng Lu (Duke) | How to "localize" the computation?]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 26 '''Thursday 3-4 pm Room 911'''<br />
| [http://eugeniacheng.com/ Eugenia Cheng] (School of the Art Institute of Chicago)<br />
| [[#Eugenia Cheng (School of the Art Institute of Chicago)| Character vs gender in mathematics and beyond ]]<br />
| Marshall / Friends of UW Madison Libraries<br />
|<br />
|-<br />
|Sept 27<br />
|<br />
|<br />
|-<br />
|Oct 4<br />
|<br />
|<br />
|-<br />
|Oct 11<br />
| Omer Mermelstein (Madison)<br />
| [[#Omer Mermelstein (Madison)| Generic flat pregeometries ]]<br />
|Andrews<br />
|<br />
|-<br />
|Oct 18<br />
| Shamgar Gurevich (Madison)<br />
| [[#Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields ]]<br />
| Marshall<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Elchanan Mossel (MIT)<br />
|Distinguished Lecture<br />
|Roch<br />
|-<br />
|Nov 8<br />
|Jose Rodriguez (UW-Madison)<br />
|[[#Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees]]<br />
|Erman<br />
|-<br />
|Nov 13 '''Wednesday 4-5pm'''<br />
|Ananth Shankar (MIT)<br />
|Exceptional splitting of abelian surfaces<br />
|-<br />
|Nov 20 '''Wednesday 4-5pm'''<br />
|Franca Hoffman (Caltech)<br />
|[[#Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis]]<br />
|Smith<br />
|-<br />
|Nov 22<br />
| Jeffrey Danciger (UT Austin)<br />
| [[#Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture"]]<br />
| Kent<br />
|-<br />
|Nov 25 '''Monday 4-5 pm Room 911'''<br />
|Tatyana Shcherbina (Princeton)<br />
| [[# Tatyana Shcherbina (Princeton)| "Random matrix theory and supersymmetry techniques"]]<br />
|Roch<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 2 '''Monday 4-5pm'''<br />
|Tingran Gao (University of Chicago)<br />
| [[#Tingran Gao (University of Chicago)| "Manifold Learning on Fibre Bundles"]]<br />
|Smith<br />
|-<br />
|Dec 4 '''Wednesday 4-5 pm'''<br />
|Andrew Zimmer<br />
| Intrinsic and extrinsic geometries in several complex variables<br />
|Gong<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 9 '''Monday 4-5 pm'''<br />
|Hui Yu (Columbia)<br />
| Singular sets in obstacle problems<br />
|Tran<br />
|-<br />
|Dec 11 '''Wednesday'''<br />
|Nick Higham (Manchester)<br />
|LAA lecture<br />
|Brualdi<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Jan 31<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 7<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 14<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 21<br />
|Shai Evra (IAS)<br />
|<br />
|Gurevich<br />
|<br />
|-<br />
|Feb 28<br />
|Brett Wick (Washington University, St. Louis)<br />
|<br />
|Seeger<br />
|-<br />
|March 6<br />
| Jessica Fintzen (Michigan)<br />
|<br />
|Marshall<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|(Moduli Spaces Conference)<br />
|<br />
|Boggess, Sankar<br />
|-<br />
|April 3<br />
|Caroline Turnage-Butterbaugh (Carleton College)<br />
|<br />
|Marshall<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|Song Sun (Berkeley)<br />
|<br />
|Huang<br />
|-<br />
|April 24<br />
|Natasa Sesum (Rutgers University)<br />
|<br />
|Angenent<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Will Sawin (Columbia)===<br />
<br />
Title: On Chowla's Conjecture over F_q[T]<br />
<br />
Abstract: The Mobius function in number theory is a sequences of 1s, <br />
-1s, and 0s, which is simple to define and closely related to the <br />
prime numbers. Its behavior seems highly random. Chowla's conjecture <br />
is one precise formalization of this randomness, and has seen recent <br />
work by Matomaki, Radziwill, Tao, and Teravainen making progress on <br />
it. In joint work with Mark Shusterman, we modify this conjecture by <br />
replacing the natural numbers parameterizing this sequence with <br />
polynomials over a finite field. Under mild conditions on the finite <br />
field, we are able to prove a strong form of this conjecture. The <br />
proof is based on taking a geometric perspective on the problem, and <br />
succeeds because we are able to simplify the geometry using a trick <br />
based on the strange properties of polynomial derivatives over finite <br />
fields.<br />
<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
<br />
===Alicia Dickenstein (Buenos Aires)===<br />
<br />
Title: Algebra and geometry in the study of enzymatic cascades<br />
<br />
Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible.<br />
I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.<br />
<br />
<br />
=== Jianfeng Lu (Duke) ===<br />
Title: How to ``localize" the computation?<br />
<br />
It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.<br />
<br />
===Eugenia Cheng (School of the Art Institute of Chicago)===<br />
<br />
Title: Character vs gender in mathematics and beyond<br />
<br />
Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.<br />
<br />
===Omer Mermelstein (Madison)===<br />
<br />
Title: Generic flat pregeometries<br />
<br />
Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure.<br />
The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.<br />
<br />
<br />
===Shamgar Gurevich (Madison)===<br />
<br />
Title: Harmonic Analysis on GL(n) over Finite Fields.<br />
<br />
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:<br />
<br />
trace(ρ(g)) / dim(ρ),<br />
<br />
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.<br />
<br />
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. <br />
<br />
This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.<br />
<br />
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)<br />
<br />
<br />
===Jose Rodriguez (UW-Madison)===<br />
Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science,<br />
engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient<br />
descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer.<br />
An alternative method is to determine the critical points of an objective function on the model.<br />
In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree).<br />
In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover,<br />
I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for<br />
determining critical points of the squared Euclidean distance function.<br />
<br />
<br />
===Ananth Shankar (MIT)===<br />
An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfaces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.<br />
<br />
===Franca Hoffman (Caltech)===<br />
<br />
Title: Gradient Flows: From PDE to Data Analysis.<br />
<br />
Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches.<br />
In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs.<br />
In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.<br />
<br />
=== Jeffrey Danciger (UT Austin) ===<br />
<br />
Title: Affine geometry and the Auslander Conjecture<br />
<br />
Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. <br />
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm&uuml;ller theory will enter the picture.<br />
<br />
=== Tatyana Shcherbina (Princeton) ===<br />
<br />
Title: Random matrix theory and supersymmetry techniques<br />
<br />
Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a special structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices<br />
whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. <br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
=== Tingran Gao (University of Chicago) ===<br />
<br />
Title: Manifold Learning on Fibre Bundles<br />
<br />
Abstract: Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.<br />
<br />
=== Andrew Zimmer (LSU) ===<br />
<br />
Title: Intrinsic and extrinsic geometries in several complex variables<br />
<br />
Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.<br />
<br />
=== Hui Yu (Columbia) ===<br />
<br />
Title: Singular sets in obstacle problems<br />
<br />
Abstract: One of the most important free boundary problems is the obstacle problem. The regularity of its free boundary has been studied for over half a century. In this talk, we review some classical results as well as exciting new developments. In particular, we discuss the recent resolution of the regularity of the singular set for the fully nonlinear obstacle problem. This talk is based on a joint work with Ovidiu Savin at Columbia University.<br />
<br />
== Future Colloquia ==<br />
[[Colloquia/Fall 2020| Fall 2020]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18457PDE Geometric Analysis seminar2019-11-20T17:19:01Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 25-27, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=1015<br />
|| Forward and Inverse Problems in Kinetic Theory <br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]<br />
| Kim and Tran<br />
|- <br />
|Nov 18<br />
| Ilyas Khan (UW Madison)<br />
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]<br />
| Kim and Tran<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|Dec 9 - Colloquium (4-5PM)<br />
| Hui Yu (Columbia)<br />
|[[#Hui Yu | TBA ]]<br />
| Tran<br />
|- <br />
|Feb. 3<br />
| Philippe LeFloch (Sorbonne Université)<br />
|[[#Speaker | TBA ]]<br />
| Feldman<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Matthew Schrecker (UW Madison)<br />
|[[#Matthew Schrecker | TBA ]]<br />
| Feldman<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Huy Nguyen (Brown)<br />
|[[#Huy Nguyen | TBA ]]<br />
| organizer<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.<br />
<br />
===Albert Ai===<br />
Title: Two dimensional gravity waves at low regularity: Energy estimates<br />
<br />
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.<br />
<br />
===Ilyas Khan===<br />
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.<br />
<br />
Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=18456Colloquia2019-11-20T17:17:28Z<p>Hung: /* Fall 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6 '''Room 911'''<br />
| Will Sawin (Columbia)<br />
| [[#Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] ]]<br />
| Marshall<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| [http://mate.dm.uba.ar/~alidick/ Alicia Dickenstein] (Buenos Aires)<br />
|[[#Alicia Dickenstein (Buenos Aires)| Algebra and geometry in the study of enzymatic cascades ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| [https://math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke)<br />
|[[#Jianfeng Lu (Duke) | How to "localize" the computation?]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 26 '''Thursday 3-4 pm Room 911'''<br />
| [http://eugeniacheng.com/ Eugenia Cheng] (School of the Art Institute of Chicago)<br />
| [[#Eugenia Cheng (School of the Art Institute of Chicago)| Character vs gender in mathematics and beyond ]]<br />
| Marshall / Friends of UW Madison Libraries<br />
|<br />
|-<br />
|Sept 27<br />
|<br />
|<br />
|-<br />
|Oct 4<br />
|<br />
|<br />
|-<br />
|Oct 11<br />
| Omer Mermelstein (Madison)<br />
| [[#Omer Mermelstein (Madison)| Generic flat pregeometries ]]<br />
|Andrews<br />
|<br />
|-<br />
|Oct 18<br />
| Shamgar Gurevich (Madison)<br />
| [[#Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields ]]<br />
| Marshall<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Elchanan Mossel (MIT)<br />
|Distinguished Lecture<br />
|Roch<br />
|-<br />
|Nov 8<br />
|Jose Rodriguez (UW-Madison)<br />
|[[#Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees]]<br />
|Erman<br />
|-<br />
|Nov 13 '''Wednesday 4-5pm'''<br />
|Ananth Shankar (MIT)<br />
|Exceptional splitting of abelian surfaces<br />
|-<br />
|Nov 20 '''Wednesday 4-5pm'''<br />
|Franca Hoffman (Caltech)<br />
|[[#Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis]]<br />
|Smith<br />
|-<br />
|Nov 22<br />
| Jeffrey Danciger (UT Austin)<br />
| [[#Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture"]]<br />
| Kent<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 2 '''Monday 4-5pm'''<br />
|Tingran Gao<br />
|<br />
|Smith<br />
|-<br />
|Dec 4 '''Wednesday 4-5 pm'''<br />
|Andrew Zimmer<br />
| Intrinsic and extrinsic geometries in several complex variables<br />
|Gong<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 9 '''Monday 4-5 pm'''<br />
|Hui Yu (Columbia)<br />
| <br />
|Tran<br />
|-<br />
|Dec 11 '''Wednesday'''<br />
|Nick Higham (Manchester)<br />
|LAA lecture<br />
|Brualdi<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Jan 31<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 7<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 14<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Feb 21<br />
|Shai Evra (IAS)<br />
|<br />
|Gurevich<br />
|<br />
|-<br />
|Feb 28<br />
|Brett Wick (Washington University, St. Louis)<br />
|<br />
|Seeger<br />
|-<br />
|March 6<br />
| Jessica Fintzen (Michigan)<br />
|<br />
|Marshall<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|(Moduli Spaces Conference)<br />
|<br />
|Boggess, Sankar<br />
|-<br />
|April 3<br />
|Caroline Turnage-Butterbaugh (Carleton College)<br />
|<br />
|Marshall<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|Song Sun (Berkeley)<br />
|<br />
|Huang<br />
|-<br />
|April 24<br />
|Natasa Sesum (Rutgers University)<br />
|<br />
|Angenent<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Will Sawin (Columbia)===<br />
<br />
Title: On Chowla's Conjecture over F_q[T]<br />
<br />
Abstract: The Mobius function in number theory is a sequences of 1s, <br />
-1s, and 0s, which is simple to define and closely related to the <br />
prime numbers. Its behavior seems highly random. Chowla's conjecture <br />
is one precise formalization of this randomness, and has seen recent <br />
work by Matomaki, Radziwill, Tao, and Teravainen making progress on <br />
it. In joint work with Mark Shusterman, we modify this conjecture by <br />
replacing the natural numbers parameterizing this sequence with <br />
polynomials over a finite field. Under mild conditions on the finite <br />
field, we are able to prove a strong form of this conjecture. The <br />
proof is based on taking a geometric perspective on the problem, and <br />
succeeds because we are able to simplify the geometry using a trick <br />
based on the strange properties of polynomial derivatives over finite <br />
fields.<br />
<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
<br />
===Alicia Dickenstein (Buenos Aires)===<br />
<br />
Title: Algebra and geometry in the study of enzymatic cascades<br />
<br />
Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible.<br />
I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.<br />
<br />
<br />
=== Jianfeng Lu (Duke) ===<br />
Title: How to ``localize" the computation?<br />
<br />
It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.<br />
<br />
===Eugenia Cheng (School of the Art Institute of Chicago)===<br />
<br />
Title: Character vs gender in mathematics and beyond<br />
<br />
Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.<br />
<br />
===Omer Mermelstein (Madison)===<br />
<br />
Title: Generic flat pregeometries<br />
<br />
Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure.<br />
The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.<br />
<br />
<br />
===Shamgar Gurevich (Madison)===<br />
<br />
Title: Harmonic Analysis on GL(n) over Finite Fields.<br />
<br />
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:<br />
<br />
trace(ρ(g)) / dim(ρ),<br />
<br />
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.<br />
<br />
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. <br />
<br />
This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.<br />
<br />
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)<br />
<br />
<br />
===Jose Rodriguez (UW-Madison)===<br />
Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science,<br />
engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient<br />
descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer.<br />
An alternative method is to determine the critical points of an objective function on the model.<br />
In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree).<br />
In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover,<br />
I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for<br />
determining critical points of the squared Euclidean distance function.<br />
<br />
<br />
===Ananth Shankar (MIT)===<br />
An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfaces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.<br />
<br />
===Franca Hoffman (Caltech)===<br />
<br />
Title: Gradient Flows: From PDE to Data Analysis.<br />
<br />
Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches.<br />
In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs.<br />
In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.<br />
<br />
=== Jeffrey Danciger (UT Austin) ===<br />
<br />
Title: Affine geometry and the Auslander Conjecture<br />
<br />
Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. <br />
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm&uuml;ller theory will enter the picture.<br />
<br />
=== Andrew Zimmer (LSU) ===<br />
<br />
Title: Intrinsic and extrinsic geometries in several complex variables<br />
<br />
Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.<br />
<br />
== Future Colloquia ==<br />
[[Colloquia/Fall 2020| Fall 2020]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18420PDE Geometric Analysis seminar2019-11-15T13:11:18Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 25-27, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=1015<br />
|| Forward and Inverse Problems in Kinetic Theory <br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]<br />
| Kim and Tran<br />
|- <br />
|Nov 18<br />
| Ilyas Khan (UW Madison)<br />
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]<br />
| Kim and Tran<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Matthew Schrecker (UW Madison)<br />
|[[#Matthew Schrecker | TBA ]]<br />
| Feldman<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Huy Nguyen (Brown)<br />
|[[#Huy Nguyen | TBA ]]<br />
| organizer<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.<br />
<br />
===Albert Ai===<br />
Title: Two dimensional gravity waves at low regularity: Energy estimates<br />
<br />
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.<br />
<br />
===Ilyas Khan===<br />
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.<br />
<br />
Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18419PDE Geometric Analysis seminar2019-11-15T13:10:16Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 25-27, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=1015<br />
|| Forward and Inverse Problems in Kinetic Theory <br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]<br />
| Kim and Tran<br />
|- <br />
|Nov 18<br />
| Ilyas Khan (UW Madison)<br />
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]<br />
| Kim and Tran<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Matthew Schrecker (UW Madison)<br />
|[[#Matthew Schrecker | TBA ]]<br />
| Feldman<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Huy Nguyen (Brown)<br />
|[[#Huy Nguyen | TBA ]]<br />
| organizer<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.<br />
<br />
===Albert Ai===<br />
Title: Two dimensional gravity waves at low regularity: Energy estimates<br />
<br />
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Applied/ACMS/Spring2020&diff=18411Applied/ACMS/Spring20202019-11-14T19:14:15Z<p>Hung: /* Spring 2020 */</p>
<hr />
<div>== Spring 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| Jan 31<br />
|[https://www.math.wisc.edu/~hung/ Hung Tran] (UW-Madison)<br />
|''[[Applied/ACMS/absS20#Hung Tran (UW-Madison)| Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel]]''<br />
| Li<br />
|-<br />
| Feb 7<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Feb 14<br />
|[http://math.mit.edu/~vadicgor/ Gorin Vadim] (UW-Madison)<br />
|''[[Applied/ACMS/absS20#Gorin Vadim (UW-Madison)|TBA, either random matrix or KPZ equation]]''<br />
| Li<br />
|-<br />
| Feb 21<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Feb 28<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Mar 6<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Mar 13<br />
|[http://www.columbia.edu/~ktm2132/ Kyle Mandli] (Columbia)<br />
|''[[Applied/ACMS/absS20#Kyle Mandli (Columbia)|TBA]]''<br />
| Wally<br />
|-<br />
| Mar 20<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Mar 27<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Apr 3<br />
|[website] (institution)<br />
|''[[Applied/ACMS/absS20#Speaker (institution)|title]]''<br />
| host<br />
|-<br />
| Apr 10<br />
|[https://www.princeton.edu/~lecoanet/ Daniel Lecoanet] (Princeton)<br />
|''[[Applied/ACMS/absS20#Daniel Lecoanet (Princeton)|TBA]]''<br />
| Wally<br />
|-<br />
| Apr 17<br />
|[https://www.ornl.gov/staff-profile/hoang-tran Hoang Tran] (Oak Ridge National Laboratory)<br />
|''[[Applied/ACMS/absS20#Hoang Tran (institution)|title]]''<br />
| Tran<br />
|-<br />
| Apr 24<br />
|[https://www.pml.unc.edu/ Pedro Saenz] (UNC)<br />
|''[[Applied/ACMS/absF19#Pedro Saenz (UNC)|TBA]]''<br />
| Spagnolie</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18346PDE Geometric Analysis seminar2019-11-06T23:29:55Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 25-27, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=1015<br />
|| Forward and Inverse Problems in Kinetic Theory <br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]<br />
| Kim and Tran<br />
|- <br />
|Nov 18<br />
| Ilyas Khan (UW Madison)<br />
|[[#Ilyas Khan | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Matthew Schrecker (UW Madison)<br />
|[[#Matthew Schrecker | TBA ]]<br />
| Feldman<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Huy Nguyen (Brown)<br />
|[[#Huy Nguyen | TBA ]]<br />
| organizer<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.<br />
<br />
===Albert Ai===<br />
Title: Two dimensional gravity waves at low regularity: Energy estimates<br />
<br />
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=18293Research at UW-Madison in DifferentialEquations2019-11-03T03:19:07Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo]<br />
<br />
Matthew Schrecker<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[http://people.math.gatech.edu/~yyao9/ Yao Yao] (VV assist prof 2012-2015) Current position: Assistant Professor, Georgia Institute of Technology (2015-)<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017) Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018) Current position: Assistant Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018) Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=18292Research at UW-Madison in DifferentialEquations2019-11-03T03:18:21Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo]<br />
[Matthew Schrecker]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[http://people.math.gatech.edu/~yyao9/ Yao Yao] (VV assist prof 2012-2015) Current position: Assistant Professor, Georgia Institute of Technology (2015-)<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017) Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018) Current position: Assistant Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018) Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18283PDE Geometric Analysis seminar2019-10-31T16:02:50Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 25-27, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=1015<br />
|| Forward and Inverse Problems in Kinetic Theory <br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Ilyas Khan (UW Madison)<br />
|[[#Ilyas Khan | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.<br />
<br />
===Albert Ai===<br />
Title: Two dimensional gravity waves at low regularity: Energy estimates<br />
<br />
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=18237Research at UW-Madison in DifferentialEquations2019-10-24T01:35:54Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18158PDE Geometric Analysis seminar2019-10-14T03:31:34Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18157PDE Geometric Analysis seminar2019-10-14T03:31:12Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|[[#Speaker | TBA ]]<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18156PDE Geometric Analysis seminar2019-10-14T03:30:50Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
|<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18155PDE Geometric Analysis seminar2019-10-14T03:30:30Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
|- <br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18154PDE Geometric Analysis seminar2019-10-14T03:30:09Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
|May 18-21<br />
| Madison Workshop in PDE 2020<br />
| Tran<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18149PDE Geometric Analysis seminar2019-10-11T22:57:19Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
===Stefania Patrizi===<br />
<br />
Title: <br />
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity<br />
<br />
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.<br />
<br />
<br />
===Claude Bardos===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=18148PDE Geometric Analysis seminar2019-10-11T22:55:02Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| <br />
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| https://www.ki-net.umd.edu/content/conf?event_id=993<br />
| | Recent progress in analytical aspects of kinetic equations and related fluid models <br />
| <br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Ian Tice (CMU)<br />
|[[#Ian Tice| TBA ]]<br />
| Kim<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).<br />
<br />
<br />
===Jin Woo Jang===<br />
<br />
Title: On a Cauchy problem for the Landau-Boltzmann equation<br />
<br />
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.<br />
<br />
<br />
=== Claude Bardos ===<br />
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture<br />
<br />
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.<br />
<br />
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.<br />
<br />
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17919PDE Geometric Analysis seminar2019-09-18T15:45:10Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17918PDE Geometric Analysis seminar2019-09-18T15:44:56Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.<br />
<br />
<br />
===Son Tu===<br />
<br />
Title: State-Constraint static Hamilton-Jacobi equations in nested domains<br />
<br />
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17917PDE Geometric Analysis seminar2019-09-18T15:42:34Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 28-29, VV901<br />
| Recent progress in analytical aspects of kinetic equations and related fluid models https://www.ki-net.umd.edu/content/conf?event_id=993<br />
|[[# | ]]<br />
| <br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Hyunju Kwon (IAS)<br />
|[[#Hyunju Kwon | TBA ]]<br />
| Kim<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17736PDE Geometric Analysis seminar2019-09-03T16:55:25Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Scott Smith===<br />
<br />
Title: Recent progress on singular, quasi-linear stochastic PDE<br />
<br />
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17735PDE Geometric Analysis seminar2019-09-03T16:54:32Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Claude Bardos (Université Paris Denis Diderot, France)<br />
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]<br />
| Li<br />
|- <br />
|Oct 28<br />
| Albert Ai (UW Madison)<br />
|[[#Albert Ai | TBA ]]<br />
| Ifrim<br />
|- <br />
|Nov 4<br />
| Yunbai Cao (UW Madison)<br />
|[[#Yunbai Cao | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|-<br />
|Nov 25<br />
| Mathew Langford (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Theodora Bourni (UT Knoxville)<br />
|[[#Speaker | TBA ]]<br />
| Angenent<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Jared Speck (Vanderbilt)<br />
|[[#Jared Speck | TBA ]]<br />
| SCHRECKER<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=17602Research at UW-Madison in DifferentialEquations2019-08-01T13:24:11Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison, 2008) Geometric Analysis, Ricci flow, Kaehler-Ricci flow, Mean Curvature flow.<br />
<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=17601Research at UW-Madison in DifferentialEquations2019-08-01T13:21:49Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
This spring the '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' will be held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison, 2008) Geometric Analysis, Ricci flow, Kaehler-Ricci flow, Mean Curvature flow<br />
<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynanamical Systems, Nonlinear Analysis<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=17600Research at UW-Madison in DifferentialEquations2019-08-01T13:21:01Z<p>Hung: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
==Previous events==<br />
<br />
This spring the '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' will be held in Madison on April 21/22 (2018).<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986)<br />
Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations<br />
<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[http://www.math.wisc.edu/~ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
<br />
<br />
==Faculty in related areas==<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison, 2008) Geometric Analysis, Ricci flow, Kaehler-Ricci flow, Mean Curvature flow<br />
<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynanamical Systems, Nonlinear Analysis<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17567Fall 2019-Spring 20202019-07-19T14:35:59Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Scott Smith (UW Madison)<br />
|[[#Scott Smith | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17503Fall 2019-Spring 20202019-07-02T08:29:57Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Son Tu (UW Madison)<br />
|[[#Son Tu | TBA ]]<br />
| Kim and Tran<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17478Fall 2019-Spring 20202019-06-03T14:11:07Z<p>Hung: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Stefania Patrizi (UT Austin)<br />
|[[#Stefania Patrizi | TBA ]]<br />
| Tran<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Yannick Sire (JHU)<br />
|[[#Yannick Sire (JHU) | TBA ]]<br />
| Tran<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17303PDE Geometric Analysis seminar2019-04-10T21:54:19Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | Speeds and Homogenization for Reaction-Diffusion Equations in Random Media ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
===Yao Yao===<br />
<br />
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations<br />
<br />
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.<br />
<br />
===Jessica Lin===<br />
<br />
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media<br />
<br />
Abstract: <br />
The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17302PDE Geometric Analysis seminar2019-04-10T21:54:01Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
===Yao Yao===<br />
<br />
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations<br />
<br />
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.<br />
<br />
===Jessica Lin===<br />
<br />
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media<br />
<br />
Abstract: <br />
The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17273PDE Geometric Analysis seminar2019-04-02T13:27:01Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
===Yao Yao===<br />
<br />
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations<br />
<br />
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17272PDE Geometric Analysis seminar2019-04-02T13:26:25Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
===Jingrui Cheng===<br />
<br />
Title: Gradient estimate for complex Monge-Ampere equations<br />
<br />
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16959PDE Geometric Analysis seminar2019-02-17T14:47:11Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16958PDE Geometric Analysis seminar2019-02-17T14:46:14Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16930PDE Geometric Analysis seminar2019-02-15T16:56:02Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16929PDE Geometric Analysis seminar2019-02-15T16:55:09Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16904PDE Geometric Analysis seminar2019-02-11T17:42:37Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16903PDE Geometric Analysis seminar2019-02-11T17:41:54Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16902PDE Geometric Analysis seminar2019-02-11T15:26:55Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16721PDE Geometric Analysis seminar2019-01-25T15:03:24Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16720PDE Geometric Analysis seminar2019-01-25T15:02:45Z<p>Hung: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16719PDE Geometric Analysis seminar2019-01-25T15:01:30Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16585PDE Geometric Analysis seminar2018-12-31T07:27:04Z<p>Hung: </p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16526PDE Geometric Analysis seminar2018-12-04T20:20:22Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18,<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD in February,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16431PDE Geometric Analysis seminar2018-11-20T19:56:58Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | TBA ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | TBA ]]<br />
| Mihaela Ifrim <br />
|- <br />
|January 28,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|-<br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD in February,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|-<br />
|Time: TBD,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16359PDE Geometric Analysis seminar2018-11-07T23:20:04Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3,<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | TBA ]]<br />
| Kim and Tran <br />
|-<br />
|December 10,<br />
|Serena Frederico (MIT)<br />
|[[#Serena Frederico | TBA ]]<br />
| Mihaela Ifrim <br />
|- <br />
|January 28,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|-<br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.</div>Hunghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16358PDE Geometric Analysis seminar2018-11-07T23:19:36Z<p>Hung: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3,<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | TBA ]]<br />
| Kim and Tran <br />
|-<br />
|December 10,<br />
|Serena Frederico (MIT)<br />
|[[#Serena Frederico | TBA ]]<br />
| Mihaela Ifrim <br />
|- <br />
|January 28,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|-<br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Tran<br />
|-<br />
|Time: TBD,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.</div>Hung