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The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
The seminar will be held  in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.


===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Fall 2018 | Tentative schedule for Fall 2018]]===
== PDE GA Seminar Schedule Spring 2018 ==




= Seminar Schedule Fall 2013 =
{| cellpadding="8"
{| cellpadding="8"
!align="left" | date   
!style="width:20%" align="left" | date   
!align="left" | speaker
!align="left" | speaker
!align="left" | title
!align="left" | title
!align="left" | host(s)
!style="width:20%" align="left" | host(s)
|-
 
|September 9
|-
|Greg Drugan (U. of Washington)
|January 29, '''3-3:50PM,  B341 VV.'''
|[[#Greg Drugan (U. of Washington) |
| Dan Knopf (UT Austin)
  Construction of immersed self-shrinkers]]
|[[#Dan Knopf Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons]]
|Angenent
| Angenent
|-
|-  
|-
|February 5,  '''3-3:50PM,  B341 VV.'''
|October 7
| Andreas Seeger (UW)
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]
|[[#Andreas Seeger |  Singular integrals and  a problem on mixing flows ]]
|[[#Guo Luo (Caltech) |
| Kim & Tran
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]
|-
|Kiselev
|February 12
|-
| Sam Krupa (UT-Austin)
|-
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
|November 18
| Lee
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]
|-
|[[#Roman Shterenberg (UAB) |
|February 19
Recent progress in multidimensional periodic and almost-periodic spectral
| Maja Taskovic (UPenn)
problems. ]]
|[[#Maja Taskovic |  Exponential tails for the non-cutoff Boltzmann equation ]]
|Kiselev
| Kim
|-
|-  
|-
|February 26
|November 25
| Ashish Kumar Pandey (UIUC)
|Myeongju Chae (Hankyong National University visiting UW)
|[[# | Instabilities in shallow water wave models  ]]
|[[#Myeongju Chae (Hankyong National University) |
| Kim & Lee
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]
|-  
|Kiselev
|March 5
|-
| Khai Nguyen (NCSU)
myeongju Chae
|[[#Khai Nguyen |  Burgers Equation with Some Nonlocal Sources ]]
|-
| Tran
|December 2
|-  
|Xiaojie Wang
|March 12
|[[#Xiaojie Wang (Stony Brook University) |
| Hongwei Gao (UCLA)
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]
|[[#Hongwei Gao | Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]
|Wang
| Tran
|-
|-  
|March 19
| Huy Nguyen (Princeton)
|[[#Huy Nguyen | Compressible fluids and active potentials ]]
| Lee
|-
|-
|December 16
|March 26
|Antonio Ache(Princeton)
|  
|[[#Antonio Ache(Princeton) |
|[[# |  Spring recess (Mar 24-Apr 1, 2018) ]]
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]
|
|Viaclovsky
|-
|-
|April 2
| In-Jee Jeong (Princeton)
|[[#In-Jee Jeong |  Singularity formation for the 3D axisymmetric Euler equations ]]
| Kim
|-
|April 9
| Jeff Calder (Minnesota)
|[[#Jeff Calder |  Nonlinear PDE continuum limits in data science and machine learning ]]
| Tran
|-
|April 21-22 (Saturday-Sunday)
| [https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]
|[[#Midwest PDE seminar |  ]]
| Angenent, Feldman, Kim, Tran.
|-
|April 25 (Wednesday)
| Hitoshi Ishii (Wasow lecture)
|[[#Hitoshi Ishii |  Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory]]
| Tran.
|}
|}


= Seminar Schedule Spring 2014 =
== Abstracts ==
{| cellpadding="8"
 
!align="left" | date 
===Dan Knopf===
!align="left" | speaker
 
!align="left" | title
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
!align="left" | host(s)
 
|-
Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis
 
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]
===Andreas Seeger===
|[[#Jean-Michel Roquejoffre (Toulouse) |
 
Front propagation in the presence of integral diffusion. ]]
Title: Singular integrals and a problem on mixing flows
|Zlatos
 
|-
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
|-
 
|February 10
===Sam Krupa===
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]
 
|[[#Myoungjean Bae (POSTECH) |
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
Free Boundary Problem related to Euler-Poisson system. ]]
 
|Feldman
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.
|-
 
|-
 
|February 24
===Maja Taskovic===
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]
 
|[[#Changhui Tan (Maryland) |
Title: Exponential tails for the non-cutoff Boltzmann equation
Global classical solution and long time behavior of macroscopic flocking models. ]]
 
|Kiselev
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel.  For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.
|-
|-
|March 3
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]
|[[#Hongjie Dong (Brown) |
Parabolic equations in time-varying domains. ]]
|Kiselev
|-
|-
|March 10
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]
|[[#Hao Jia (University of Chicago) |
Long time dynamics of energy critical defocusing wave equation with
radial potential in 3+1 dimensions. ]]
|Kiselev
|-
|-
|March 31
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]
|[[#Alexander Pushnitski (King's College) |
An inverse spectral problem for Hankel operators. ]]
|Kiselev
|-
|-
|April 21
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]
|[[#Ronghua Pan (Georgia Tech) |
TBA. ]]
|Kiselev
|-
|}


= Seminar Schedule Fall 2014 =
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.
{| cellpadding="8"
!align="left" | date 
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|September 22 (joint with Analysis Seminar)
|Steven Hofmann (U. of Missouri)
|[[#Steven Hofmann (U. of Missouri) |
TBA]]
|Seeger
|-
|}


= Abstracts =


===Greg Drugan (U. of Washington)===
===Ashish Kumar Pandey===
''Construction of immersed self-shrinkers''


Abstract: We describe a procedure for constructing immersed
Title: Instabilities in shallow water wave models
self-shrinking solutions to mean curvature flow.
The self-shrinkers we construct have a rotational symmetry, and
the construction involves a detailed study of geodesics in the
upper-half plane with a conformal metric.
This is a joint work with Stephen Kleene.


===Guo Luo (Caltech)===
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''


Abstract:
Whether the 3D incompressible Euler equations can develop a singularity in
finite time from smooth initial data is one of the most challenging problems in
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this
long-standing open question from a numerical point of view, by presenting a class of
potentially singular solutions to the Euler equations computed in axisymmetric
geometries. The solutions satisfy a periodic boundary condition along the axial direction
and no-flow boundary condition on the solid wall. The equations are discretized in space
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially
designed adaptive (moving) meshes that are dynamically adjusted to the evolving
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also
suggests that the blowing-up solution develops a self-similar structure near the point of
the singularity, as the singularity time is approached.


===Xiaojie Wang(Stony Brook)===
===Khai Nguyen===
''Uniqueness of Ricci flow solutions on noncompact manifolds''


Abstract:
Title: Burgers Equation with Some Nonlocal Sources
Ricci flow is an important evolution equation of Riemannian metrics.
Since it was  introduced by R. Hamilton in 1982, it has greatly changed the  landscape of riemannian geometry.  One of the fundamental question about ricci flow is when is its  solution to initial value problem unique. On compact manifold,  with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when  further restrictions are imposed to the solution.  In this talk, we will discuss various conditions that guarantee the  uniqueness. In particular, we will discuss in  details with the following uniqueness result.    Let $(M,g)$ be a complete noncompact non-collapsing  $n$-dimensional riemannian  manifold, whose complex sectional curvature is bounded from below  and scalar curvature is bounded from above. Then ricci  flow with above  as its initial data, on $M\times [0,\epsilon]$ for some  $\epsilon>0$, has at most one solution in the class of  complete riemannian metric with complex sectional curvature bounded  from below.


===Roman Shterenberg(UAB)===
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency.  This talk  will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations.  Some open questions will be discussed.
''Recent progress in multidimensional periodic and almost-periodic spectral
problems''


Abstract: We present a review of the results in multidimensional periodic
===Hongwei Gao===  
and almost-periodic spectral problems. We discuss some recent progress and
old/new ideas used in the constructions. The talk is mostly based on the
joint works with Yu. Karpeshina and L. Parnovski.


===Antonio Ache(Princeton)===
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations
''Ricci Curvature and the manifold learning problem''


Abstract: In the first half of this talk we will review several notions of coarse or weak
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
motivation for developing a method to estimate the Ricci curvature of a an embedded
submaifold of Euclidean space from a point cloud which has applications to the Manifold
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space
from point clouds. This is joint work with Micah Warren.


===Jean-Michel Roquejoffre (Toulouse)===
===Huy Nguyen===
''Front propagation in the presence of integral diffusion''


Abstract: In many reaction-diffusion equations, where diffusion is
Title : Compressible fluids and active potentials
given by a second order elliptic operator, the solutions
will exhibit spatial transitions whose velocity is asymptotically
linear in time. The situation can be different when the diffusion is of the
integral type, the most basic example being the fractional Laplacian:
the velocity can be time-exponential. We will explain why, and
discuss several situations where this type of fast propagation
occurs.


===Myoungjean Bae (POSTECH)===
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.
''Free Boundary Problem related to Euler-Poisson system''


One dimensional analysis of Euler-Poisson system shows that when incoming
===In-Jee Jeong===
supersonic flow is fixed, transonic shock can be represented as a monotone
function of exit pressure. From this observation, we expect well-posedness
of transonic shock problem for Euler-Poisson system when exit pressure is
prescribed in a proper range. In this talk, I will present recent progress
on transonic shock problem for Euler-Poisson system, which is formulated
as a free boundary problem with mixed type PDE system.
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU)
and Jingjing Xiao(CUHK).


===Changhui Tan (University of Maryland)===
Title: Singularity formation for the 3D axisymmetric Euler equations
''Global classical solution and long time behavior of macroscopic flocking models''


Abstract: Self-organized behaviors are very common in nature and human societies.
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains  $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2  \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.
One widely discussed example is the flocking phenomenon which describes
animal groups emerging towards the same direction. Several models such
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing
flocking behaviors. In this talk, we will discuss macroscopic representation
of flocking models. These systems can be interpreted as compressible Eulerian
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set
of initial conditions will lead to a finite time break down of the system. This
is a joint work with Eitan Tadmor.


===Hongjie Dong (Brown University)===
===Jeff Calder===
''Parabolic equations in time-varying domains''


Abstract: I will present a recent result on the Dirichlet boundary value
Title: Nonlinear PDE continuum limits in data science and machine learning
problem for parabolic equations in time-varying domains. The equations are
in either divergence or non-divergence form with boundary blowup low-order
coefficients. The domains satisfy an exterior measure condition.


===Hao Jia (University of Chicago)===
Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.
''Long time dynamics of energy critical defocusing wave equation with
radial potential in 3+1 dimensions.''


Abstract: We consider the long term dynamics of radial solution to the
===Hitoshi Ishii===
above mentioned equation. For general potential, the equation can have a
unique positive ground state and a number of excited states. One can expect
that some solutions might stay for very long time near excited states
before settling down to an excited state of lower energy or the ground
state. Thus the detailed dynamics can be extremely complicated. However
using the ``channel of energy" inequality discovered by T.Duyckaerts,
C.Kenig and F.Merle, we can show for generic potential, any radial solution
is asymptotically the sum of a free radiation and a steady state as time
goes to infinity. This provides another example of the power of ``channel
of energy" inequality and the method of profile decompositions. I will
explain the basic tools in some detail. Joint work with Baoping Liu and
Guixiang Xu.


===Alexander Pushnitski (King's College)===
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory
''An inverse spectral problem for Hankel operators''


Abstract:
Abstract:   In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.
I will discuss an inverse spectral problem for a certain class of Hankel
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a
step towards description of evolution in a model integrable non-dispersive
equation. Several features of this inverse problem make it strikingly (and somewhat
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will
describe the available results for Hankel operators, emphasizing this similarity.
This is joint work with Patrick Gerard (Orsay).

Revision as of 00:34, 19 April 2018

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Tentative schedule for Fall 2018

PDE GA Seminar Schedule Spring 2018

date speaker title host(s)
January 29, 3-3:50PM, B341 VV. Dan Knopf (UT Austin) Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons Angenent
February 5, 3-3:50PM, B341 VV. Andreas Seeger (UW) Singular integrals and a problem on mixing flows Kim & Tran
February 12 Sam Krupa (UT-Austin) Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case Lee
February 19 Maja Taskovic (UPenn) Exponential tails for the non-cutoff Boltzmann equation Kim
February 26 Ashish Kumar Pandey (UIUC) Instabilities in shallow water wave models Kim & Lee
March 5 Khai Nguyen (NCSU) Burgers Equation with Some Nonlocal Sources Tran
March 12 Hongwei Gao (UCLA) Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations Tran
March 19 Huy Nguyen (Princeton) Compressible fluids and active potentials Lee
March 26 Spring recess (Mar 24-Apr 1, 2018)
April 2 In-Jee Jeong (Princeton) Singularity formation for the 3D axisymmetric Euler equations Kim
April 9 Jeff Calder (Minnesota) Nonlinear PDE continuum limits in data science and machine learning Tran
April 21-22 (Saturday-Sunday) Midwest PDE seminar Angenent, Feldman, Kim, Tran.
April 25 (Wednesday) Hitoshi Ishii (Wasow lecture) Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory Tran.

Abstracts

Dan Knopf

Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons

Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.

Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

Sam Krupa

Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case

Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.


Maja Taskovic

Title: Exponential tails for the non-cutoff Boltzmann equation

Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.

We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.


Ashish Kumar Pandey

Title: Instabilities in shallow water wave models

Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.


Khai Nguyen

Title: Burgers Equation with Some Nonlocal Sources

Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.

Hongwei Gao

Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations

Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.

Huy Nguyen

Title : Compressible fluids and active potentials

Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.

In-Jee Jeong

Title: Singularity formation for the 3D axisymmetric Euler equations

Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.

Jeff Calder

Title: Nonlinear PDE continuum limits in data science and machine learning

Abstract: We will present some recent results on PDE continuum limits for (random) discrete problems in data science and machine learning. All of the problems satisfy a type of discrete comparison/maximum principle and so the continuum PDEs are properly interpreted in the viscosity sense. We will present results for nondominated sorting, convex hull peeling, and graph-based semi-supervised learning. Nondominated sorting is an algorithm for arranging points in Euclidean space into layers by repeatedly peeling away coordinatewise minimal points, and the continuum PDE turns out to be a Hamilton-Jacobi equation. Convex hull peeling is used to order data by repeatedly peeling the vertices of the convex hull, and the continuum limit is motion by a power of Gauss curvature. Finally, a recently proposed class of graph-based learning problems have PDE continuum limits corresponding to weighted p-Laplace equations. In each case the continuum PDEs provide insights into the data science/engineering problems, and suggest avenues for fast approximate algorithms based on the PDE interpretations.

Hitoshi Ishii

Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory

Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.