Difference between revisions of "Previous PDE/GA seminars"
Line 441: | Line 441: | ||
= Abstracts = | = Abstracts = | ||
+ | |||
+ | |||
+ | ===Greg Drugan (U. of Washington)=== | ||
+ | ''Construction of immersed self-shrinkers'' | ||
+ | |||
+ | Abstract: We describe a procedure for constructing immersed | ||
+ | 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)=== | ||
+ | ''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)=== | ||
+ | ''Uniqueness of Ricci flow solutions on noncompact manifolds'' | ||
+ | |||
+ | Abstract: | ||
+ | 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)=== | ||
+ | ''Recent progress in multidimensional periodic and almost-periodic spectral | ||
+ | problems'' | ||
+ | |||
+ | Abstract: We present a review of the results in multidimensional periodic | ||
+ | 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)=== | ||
+ | ''Ricci Curvature and the manifold learning problem'' | ||
+ | |||
+ | Abstract: In the first half of this talk we will review several notions of coarse or weak | ||
+ | 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)=== | ||
+ | ''Front propagation in the presence of integral diffusion'' | ||
+ | |||
+ | Abstract: In many reaction-diffusion equations, where diffusion is | ||
+ | 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)=== | ||
+ | ''Free Boundary Problem related to Euler-Poisson system'' | ||
+ | |||
+ | One dimensional analysis of Euler-Poisson system shows that when incoming | ||
+ | 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)=== | ||
+ | ''Global classical solution and long time behavior of macroscopic flocking models'' | ||
+ | |||
+ | Abstract: Self-organized behaviors are very common in nature and human societies. | ||
+ | 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)=== | ||
+ | ''Parabolic equations in time-varying domains'' | ||
+ | |||
+ | Abstract: I will present a recent result on the Dirichlet boundary value | ||
+ | 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)=== | ||
+ | ''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 | ||
+ | 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)=== | ||
+ | ''An inverse spectral problem for Hankel operators'' | ||
+ | |||
+ | Abstract: | ||
+ | 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). | ||
+ | |||
+ | ===Ronghua Pan (Georgia Tech)=== | ||
+ | ''Compressible Navier-Stokes-Fourier system with temperature dependent dissipation'' | ||
+ | |||
+ | Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and | ||
+ | heat conductivity coefficients in compressible fluids depend on absolute temperature | ||
+ | through power laws. The mathematical theory on the well-posedness and regularity on this | ||
+ | setting is widely open. I will report some recent progress on this direction, | ||
+ | with emphasis on the lower bound of temperature, and global existence of | ||
+ | solutions in one or multiple dimensions. The relation between thermodynamics laws | ||
+ | and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works | ||
+ | with Junxiong Jia and Weizhe Zhang. | ||
===Myoungjean Bae (POSTECH)=== | ===Myoungjean Bae (POSTECH)=== |
Revision as of 09:51, 7 September 2014
Click here for the current PDE and Geometric Analysis seminar schedule.
Contents
- 1 Seminar Schedule Fall 2013
- 2 Seminar Schedule Spring 2014
- 3 Seminar Schedule Spring 2013
- 4 Seminar Schedule Fall 2012
- 5 Seminar Schedule Spring 2012
- 6 Seminar Schedule Fall 2011
- 7 Seminar Schedule Spring 2011
- 8 Seminar Schedule Fall 2010
- 9 Abstracts
- 9.1 Greg Drugan (U. of Washington)
- 9.2 Guo Luo (Caltech)
- 9.3 Xiaojie Wang(Stony Brook)
- 9.4 Roman Shterenberg(UAB)
- 9.5 Antonio Ache(Princeton)
- 9.6 Jean-Michel Roquejoffre (Toulouse)
- 9.7 Myoungjean Bae (POSTECH)
- 9.8 Changhui Tan (University of Maryland)
- 9.9 Hongjie Dong (Brown University)
- 9.10 Hao Jia (University of Chicago)
- 9.11 Alexander Pushnitski (King's College)
- 9.12 Ronghua Pan (Georgia Tech)
- 9.13 Myoungjean Bae (POSTECH)
- 9.14 Mike Cullen (Met. Office, UK)
- 9.15 Mohammad Ghomi(Math. Georgia Tech)
- 9.16 Wei Xiang (Oxford)
- 9.17 Adrian Tudorascu (West Virginia University)
- 9.18 Diego Cordoba (Madrid)
- 9.19 Fausto Ferrari (Bologna)
- 9.20 Arshak Petrosyan (Purdue)
- 9.21 Changyou Wang (U. of Kentucky)
- 9.22 Philippe LeFloch (Paris VI)
- 9.23 Irina Mitrea
- 9.24 Panagiota Daskalopoulos (Columbia U)
- 9.25 Maria Gualdani (UT Austin)
- 9.26 Hiroshi Matano (Tokyo University)
- 9.27 Ian Tice (Brown University)
- 9.28 Hoai Minh Nguyen (NYU-Courant Institute)
- 9.29 Bing Wang (Princeton)
- 9.30 Francois Hamel (Marseille)
- 9.31 Juraj Foldes (Vanderbilt)
- 9.32 Alexey Cheskidov (UIC)
- 9.33 Takis Souganidis (Chicago)
- 9.34 Scott Armstrong (UW-Madison)
- 9.35 Russell Schwab (Carnegie Mellon)
- 9.36 Valentin Ovsienko (University of Lyon)
- 9.37 Adrian Tudorascu (West Virginia University)
- 9.38 James Nolen (Duke)
- 9.39 Betsy Stovall (UCLA)
- 9.40 Charles Smart (MIT)
- 9.41 Yao Yao (UCLA)
- 9.42 Xuan Hien Nguyen (Iowa State)
- 9.43 Nestor Guillen (UCLA)
- 9.44 Charles Smart (MIT)
- 9.45 Vlad Vicol (University of Chicago)
- 9.46 Jiahong Wu (Oklahoma State)
- 9.47 Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)
- 9.48 Gui-Qiang Chen (Oxford)
- 9.49 Jacob Glenn-Levin (UT Austin)
- 9.50 Bing Wang (UW Madison)
- 9.51 Peter Polacik (University of Minnesota)
- 9.52 Kyudong Choi (UW Madison)
- 9.53 Yao Yao (UW Madison)
- 9.54 Myoungjean Bae (POSTECH)
Seminar Schedule Fall 2013
myeongju Chaedate | speaker | title | host(s) |
---|---|---|---|
September 9 | Greg Drugan (U. of Washington) |
Construction of immersed self-shrinkers |
Angenent |
October 7 | Guo Luo (Caltech) |
Potentially Singular Solutions of the 3D Incompressible Euler Equations. |
Kiselev |
November 18 | Roman Shterenberg (UAB) |
Recent progress in multidimensional periodic and almost-periodic spectral problems. |
Kiselev |
November 25 | Myeongju Chae (Hankyong National University visiting UW) | Kiselev | |
December 2 | Xiaojie Wang | Wang | |
December 16 | Antonio Ache(Princeton) | Viaclovsky |
Seminar Schedule Spring 2014
date | speaker | title | host(s) |
---|---|---|---|
January 14 at 4pm in B139 (TUESDAY), joint with Analysis | Jean-Michel Roquejoffre (Toulouse) | Zlatos | |
February 10 | Myoungjean Bae (POSTECH) | Feldman | |
February 24 | Changhui Tan (Maryland) |
Global classical solution and long time behavior of macroscopic flocking models. |
Kiselev |
March 3 | Hongjie Dong (Brown) | Kiselev | |
March 10 | Hao Jia (University of Chicago) | Kiselev | |
March 31 | Alexander Pushnitski (King's College London) | Kiselev | |
April 21 | Ronghua Pan (Georgia Tech) |
Compressible Navier-Stokes-Fourier system with temperature dependent dissipation. |
Kiselev |
Seminar Schedule Spring 2013
date | speaker | title | host(s) |
---|---|---|---|
February 4 | Myoungjean Bae (POSTECH) |
Transonic shocks for Euler-Poisson system and related problems |
Feldman |
February 18 | Mike Cullen (Met. Office, UK) |
Modelling the uncertainty in predicting large-scale atmospheric circulations. |
Feldman |
March 18 | Mohammad Ghomi(Math. Georgia Tech) | Angenent | |
April 8 | Wei Xiang (Oxford) |
Shock Diffraction Problem to the Two Dimensional Nonlinear Wave System and Potential Flow Equation. |
Feldman |
Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room) | Adrian Tudorascu (West Virginia University) | One-dimensional pressureless | Feldman |
May 6 | Diego Cordoba (Madrid) | Kiselev |
Seminar Schedule Fall 2012
date | speaker | title | host(s) |
---|---|---|---|
September 17 | Bing Wang (UW Madison) |
On the regularity of limit space |
local |
October 15 | Peter Polacik (University of Minnesota) |
Exponential separation between positive and sign-changing solutions and its applications |
Zlatos |
November 26 | Kyudong Choi (UW Madison) |
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations |
local |
December 10 | Yao Yao (UW Madison) |
Confinement for nonlocal interaction equation with repulsive-attractive kernels |
local |
Seminar Schedule Spring 2012
date | speaker | title | host(s) |
---|---|---|---|
Feb 6 | Yao Yao (UCLA) |
Degenerate diffusion with nonlocal aggregation: behavior of solutions |
Kiselev |
March 12 | Xuan Hien Nguyen (Iowa State) |
Gluing constructions for solitons and self-shrinkers under mean curvature flow |
Angenent |
March 21(Wednesday!), Room 901 Van Vleck | Nestor Guillen (UCLA) | Feldman | |
March 26 | Vlad Vicol (University of Chicago) |
Shape dependent maximum principles and applications |
Kiselev |
April 9 | Charles Smart (MIT) | Seeger | |
April 16 | Jiahong Wu (Oklahoma) |
The 2D Boussinesq equations with partial dissipation |
Kiselev |
April 23 | Joana Oliveira dos Santos Amorim (Universite Paris Dauphine) |
A geometric look on Aubry-Mather theory and a theorem of Birkhoff |
Bolotin |
April 27 (Colloquium. Friday at 4pm, in Van Vleck B239) | Gui-Qiang Chen (Oxford) |
Nonlinear Partial Differential Equations of Mixed Type |
Feldman |
May 14 | Jacob Glenn-Levin (UT Austin) |
Incompressible Boussinesq equations in borderline Besov spaces |
Kiselev |
Seminar Schedule Fall 2011
date | speaker | title | host(s) |
---|---|---|---|
Oct 3 | Takis Souganidis (Chicago) |
Stochastic homogenization of the G-equation |
Armstrong |
Oct 10 | Scott Armstrong (UW-Madison) |
Partial regularity for fully nonlinear elliptic equations |
Local speaker |
Oct 17 | Russell Schwab (Carnegie Mellon) |
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients) |
Armstrong |
October 24 ( with Geometry/Topology seminar) | Valentin Ovsienko (University of Lyon) | Marí Beffa | |
Oct 31 | Adrian Tudorascu (West Virginia University) |
Weak Lagrangian solutions for the Semi-Geostrophic system in physical space |
Feldman |
Nov 7 | James Nolen (Duke) | Armstrong | |
Nov 21 (Joint with Analysis seminar) | Betsy Stovall (UCLA) |
Scattering for the cubic Klein--Gordon equation in two dimensions |
Seeger |
Dec 5 | Charles Smart (MIT) |
Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian |
Armstrong |
Seminar Schedule Spring 2011
date | speaker | title | host(s) |
---|---|---|---|
Jan 24 | Bing Wang (Princeton) |
The Kaehler Ricci flow on Fano manifold |
Viaclovsky |
Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis) | Francois Hamel (Marseille) |
Optimization of eigenvalues of non-symmetric elliptic operators |
Zlatos |
Mar 28 | Juraj Foldes (Vanderbilt) |
Symmetry properties of parabolic problems and their applications |
Zlatos |
Apr 11 | Alexey Cheskidov (UIC) |
Navier-Stokes and Euler equations: a unified approach to the problem of blow-up |
Kiselev |
Date TBA | Mikhail Feldman (UW Madison) | TBA | Local speaker |
Date TBA | Sigurd Angenent (UW Madison) | TBA | Local speaker |
Seminar Schedule Fall 2010
date | speaker | title | host(s) |
---|---|---|---|
Sept 13 | Fausto Ferrari (Bologna) |
Semilinear PDEs and some symmetry properties of stable solutions |
Feldman |
Sept 27 | Arshak Petrosyan (Purdue) | Feldman | |
Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. Special day, time & room. | Changyou Wang (U. of Kentucky) | Feldman | |
Oct 11 | Philippe LeFloch (Paris VI) |
Kinetic relations for undercompressive shock waves and propagating phase boundaries |
Feldman |
Oct 29 Friday 2:30pm, Room: B115 Van Vleck. Special day, time & room. | Irina Mitrea (IMA) |
Boundary Value Problems for Higher Order Differential Operators |
WiMaW |
Nov 1 | Panagiota Daskalopoulos (Columbia U) | Feldman | |
Nov 8 | Maria Gualdani (UT Austin) | Feldman | |
Nov 18 Thursday 1:20pm Room: 901 Van Vleck Special day & time. | Hiroshi Matano (Tokyo University) |
Traveling waves in a sawtoothed cylinder and their homogenization limit |
Angenent & Rabinowitz |
Nov 29 | Ian Tice (Brown University) |
Global well-posedness and decay for the viscous surface wave problem without surface tension |
Feldman |
Dec. 8 Wed 2:25pm, Room: 901 Van Vleck. Special day, time & room. | Hoai Minh Nguyen (NYU-Courant Institute) | Feldman |
Abstracts
Greg Drugan (U. of Washington)
Construction of immersed self-shrinkers
Abstract: We describe a procedure for constructing immersed 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)
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)
Uniqueness of Ricci flow solutions on noncompact manifolds
Abstract: 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)
Recent progress in multidimensional periodic and almost-periodic spectral problems
Abstract: We present a review of the results in multidimensional periodic 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)
Ricci Curvature and the manifold learning problem
Abstract: In the first half of this talk we will review several notions of coarse or weak 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)
Front propagation in the presence of integral diffusion
Abstract: In many reaction-diffusion equations, where diffusion is 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)
Free Boundary Problem related to Euler-Poisson system
One dimensional analysis of Euler-Poisson system shows that when incoming 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)
Global classical solution and long time behavior of macroscopic flocking models
Abstract: Self-organized behaviors are very common in nature and human societies. 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)
Parabolic equations in time-varying domains
Abstract: I will present a recent result on the Dirichlet boundary value 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)
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 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)
An inverse spectral problem for Hankel operators
Abstract: 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).
Ronghua Pan (Georgia Tech)
Compressible Navier-Stokes-Fourier system with temperature dependent dissipation
Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and heat conductivity coefficients in compressible fluids depend on absolute temperature through power laws. The mathematical theory on the well-posedness and regularity on this setting is widely open. I will report some recent progress on this direction, with emphasis on the lower bound of temperature, and global existence of solutions in one or multiple dimensions. The relation between thermodynamics laws and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works with Junxiong Jia and Weizhe Zhang.
Myoungjean Bae (POSTECH)
Transonic shocks for Euler-Poisson system and related problems
Abstract: Euler-Poisson system models various physical phenomena including the propagation of electrons in submicron semiconductor devices and plasmas, and the biological transport of ions for channel proteins. I will explain difference between Euler system and Euler-Poisson system and mathematical difficulties arising due to this difference. And, recent results about subsonic flow and transonic flow for Euler-Poisson system will be presented. This talk is based on collaboration with Ben Duan and Chunjing Xie.
Mike Cullen (Met. Office, UK)
Modelling the uncertainty in predicting large-scale atmospheric circulations
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.
Mohammad Ghomi(Math. Georgia Tech)
Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".
Wei Xiang (Oxford)
Abstract: The vertical shock which initially separates two piecewise constant Riemann data, passes the wedge from left to right, then shock diffraction phenomena will occur and the incident shock becomes a transonic shock. Here we study this problem on nonlinear wave system as well as on potential flow equations. The existence and the optimal regularity across sonic circle of the solutions to this problem is established. The comparison of these two systems is discussed, and some related open problems are proposed.
Adrian Tudorascu (West Virginia University)
Abstract: This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).
Diego Cordoba (Madrid)
Abstract: We consider the evolution of an interface generated between two immiscible, incompressible and irrotational fluids. Specifically we study the Muskat equation (the interface between oil and water in sand) and water wave equation (interface between water and vacuum). For both equations we will study well-posedness and the existence of smooth initial data for which the smoothness of the interface breaks down in finite time. We will also discuss some open problems.
Fausto Ferrari (Bologna)
Semilinear PDEs and some symmetry properties of stable solutions
I will deal with stable solutions of semilinear elliptic PDE's and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.
Arshak Petrosyan (Purdue)
Nonuniqueness in a free boundary problem from combustion
We consider a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. V ́azquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion.
This is a joint work with Aaron Yip.
Changyou Wang (U. of Kentucky)
Phase transition for higher dimensional wells
For a potential function [math]F[/math] that has two global minimum sets consisting of two compact connected Riemannian submanifolds in [math]\mathbb{R}^k[/math], we consider the singular perturbation problem:
Minimizing [math]\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)[/math] under given Dirichlet boundary data.
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic, as the parameter [math]\epsilon[/math] tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.
Philippe LeFloch (Paris VI)
Kinetic relations for undercompressive shock waves and propagating phase boundaries
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics. In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.
Irina Mitrea
Boundary Value Problems for Higher Order Differential Operators
As is well known, many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator L in a domain D.
When L is a differential operator of second order a variety of tools are available for dealing with such problems including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. The situation when the differential operator has higher order (as is the case for instance with anisotropic plate bending when one deals with fourth order) stands in sharp contrast with this as only fewer options could be successfully implemented. Alberto Calderon, one of the founders of the modern theory of Singular Integral Operators, has advocated in the seventies the use of layer potentials for the treatment of higher order elliptic boundary value problems. While the layer potential method has proved to be tremendously successful in the treatment of second order problems, this approach is insufficiently developed to deal with the intricacies of the theory of higher order operators. In fact, it is largely absent from the literature dealing with such problems.
In this talk I will discuss recent progress in developing a multiple layer potential approach for the treatment of boundary value problems associated with higher order elliptic differential operators. This is done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory.
Panagiota Daskalopoulos (Columbia U)
Ancient solutions to geometric flows
We will discuss the clasification of ancient solutions to nonlinear geometric flows. It is well known that ancient solutions appear as blow up limits at a finite time singularity of the flow. Special emphasis will be given to the 2-dimensional Ricci flow. In this case we will show that ancient compact solution is either the Einstein (trivial) or one of the King-Rosenau solutions.
Maria Gualdani (UT Austin)
A nonlinear diffusion model in mean-field games
We present an overview of mean-field games theory and show recent results on a free boundary value problem, which models price formation dynamics. In such model, the price is formed through a game among infinite number of agents. Existence and regularity results, as well as linear stability, will be shown.
Hiroshi Matano (Tokyo University)
Traveling waves in a sawtoothed cylinder and their homogenization limit
My talk is concerned with a curvature-dependent motion of plane curves in a two-dimensional cylinder with spatially undulating boundary. In other words, the boundary has many bumps and we assume that the bumps are aligned in a spatially recurrent manner.
The goal is to study how the average speed of the traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the boundary undulation becomes finer and finer, and determine the homogenization limit of the average speed and the limit profile of the traveling waves. Quite surprisingly, this homogenized speed depends only on the maximal opening angles of the domain boundary and no other geometrical features are relevant.
Next we consider the special case where the boundary undulation is quasi-periodic with m independent frequencies. We show that the rate of convergence to the homogenization limit depends on this number m.
This is joint work with Bendong Lou and Ken-Ichi Nakamura.
Ian Tice (Brown University)
Global well-posedness and decay for the viscous surface wave problem without surface tension
We study the incompressible, gravity-driven Navier-Stokes equations in three dimensional domains with free upper boundaries and fixed lower boundaries, in both the horizontally periodic and non-periodic settings. The effect of surface tension is not included. We employ a novel two-tier nonlinear energy method that couples the boundedness of certain high-regularity norms to the algebraic decay of lower-regularity norms. The algebraic decay allows us to balance the growth of the highest order derivatives of the free surface function, which then allows us to derive a priori estimates for solutions. We then prove local well-posedness in our energy space, which yields global well-posedness and decay. The novel LWP theory is established through the study of the linear Stokes problem in moving domains. This is joint work with Yan Guo.
Hoai Minh Nguyen (NYU-Courant Institute)
Cloaking via change of variables for the Helmholtz equation
A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equation. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. If time permits, I will mention some results related to the wave equation.
Bing Wang (Princeton)
The Kaehler Ricci flow on Fano manifold
We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big [math]\alpha_{\nu, 1}[/math] or [math]\alpha_{\nu, 2}[/math] (Tian's invariants). Our method also works for 2-dimensional Fano orbifolds. Since Tian's invariants can be calculated by algebraic geometry method, our convergence theorem implies that one can find new Kaehler Einstein metrics on orbifolds by calculating Tian's invariants. An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.
Francois Hamel (Marseille)
Optimization of eigenvalues of non-symmetric elliptic operators
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of [math]R^n[/math]. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
Juraj Foldes (Vanderbilt)
Symmetry properties of parabolic problems and their applications
Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.
Alexey Cheskidov (UIC)
Navier-Stokes and Euler equations: a unified approach to the problem of blow-up
The problems of blow-up for Navier-Stokes and Euler equations have been extensively studied for decades using different techniques. Motivated by Kolmogorov's theory of turbulence, we present a new unified approach to the blow-up problem for the equations of incompressible fluid motion. In particular, we present a new regularity criterion which is weaker than the Beale-Kato-Majda condition in the inviscid case, and weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.
Takis Souganidis (Chicago)
Stochastic homogenization of the G-equation
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.
Scott Armstrong (UW-Madison)
Partial regularity for fully nonlinear elliptic equations
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.
Russell Schwab (Carnegie Mellon)
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])
Valentin Ovsienko (University of Lyon)
The pentagram map and generalized friezes of Coxeter
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map. In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.
Adrian Tudorascu (West Virginia University)
Weak Lagrangian solutions for the Semi-Geostrophic system in physical space
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.
James Nolen (Duke)
Normal approximation for a random elliptic PDE
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.
Betsy Stovall (UCLA)
We will discuss recent work concerning the cubic Klein--Gordon equation u_{tt} - \Delta u + u \pm u^3 = 0 in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain questions arising in harmonic analysis.
This is joint work with Rowan Killip and Monica Visan.
Charles Smart (MIT)
Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.
Yao Yao (UCLA)
Degenerate diffusion with nonlocal aggregation: behavior of solutions
The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow up criteria are well known, the asymptotic behaviors of solutions are not completely clear. In this talk I will present some results on the asymptotic behavior of solutions when there is global existence. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions. This is a joint work with Inwon Kim.
Xuan Hien Nguyen (Iowa State)
Gluing constructions for solitons and self-shrinkers under mean curvature flow
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.
Nestor Guillen (UCLA)
We consider the Monge-Kantorovich problem, which consists in transporting a given measure into another "target" measure in a way that minimizes the total cost of moving each unit of mass to its new location. When the transport cost is given by the square of the distance between two points, the optimal map is given by a convex potential which solves the Monge-Ampère equation, in general, the solution is given by what is called a c-convex potential. In recent work with Jun Kitagawa, we prove local Holder estimates of optimal transport maps for more general cost functions satisfying a "synthetic" MTW condition, in particular, the proof does not really use the C^4 assumption made in all previous works. A similar result was recently obtained by Figalli, Kim and McCann using different methods and assuming strict convexity of the target.
Charles Smart (MIT)
PDE methods for the Abelian sandpile
Abstract: The Abelian sandpile growth model is a deterministic diffusion process for chips placed on the $d$-dimensional integer lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar fractal limit when begun from increasingly large stacks of chips at the origin. This behavior defied explanation for many years until viscosity solution theory offered a new perspective. This is joint work with Lionel Levine and Wesley Pegden.
Vlad Vicol (University of Chicago)
Title: Shape dependent maximum principles and applications
Abstract: We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with P. Constantin.
Jiahong Wu (Oklahoma State)
"The 2D Boussinesq equations with partial dissipation"
The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial dissipation has attracted considerable attention in the last few years. In this talk we will summarize recent results on various cases of partial dissipation, present the work of Cao and Wu on the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on the critical Boussinesq equations with a logarithmically singular velocity.
Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)
"A geometric look on Aubry-Mather theory and a theorem of Birkhoff"
Given a Tonelli Hamiltonian $H:T^*M \lto \Rm$ in the cotangent bundle of a compact manifold $M$, we can study its dynamic using the Aubry and Ma\~n\'e sets defined by Mather. In this talk we will explain their importance and give a new geometric definition which allows us to understand their property of symplectic invariance. Moreover, using this geometric definition, we will show that an exact Lipchitz Lagrangian manifold isotopic to a graph which is invariant by the flow of a Tonelli Hamiltonian is itself a graph. This result, in its smooth form, was a conjecture of Birkhoff.
Gui-Qiang Chen (Oxford)
"Nonlinear Partial Differential Equations of Mixed Type"
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed. This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang.
Jacob Glenn-Levin (UT Austin)
We consider the Boussinesq equations, which may be thought of as inhomogeneous, incompressible Euler equations, where the inhomogeneous term is a scalar quantity, typically density or temperature, governed by a convection-diffusion equation. I will discuss local- and global-in-time well-posedness results for the incompressible 2D Boussinesq equations, assuming the density equation has nonzero diffusion and that the initial data belongs in a Besov-type space.
Bing Wang (UW Madison)
On the regularity of limit space
This is a joint work with Gang Tian. In this talk, we will discuss how to improve regularity of the limit space by Ricci flow. We study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L1-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of K¨ahler manifolds.
Peter Polacik (University of Minnesota)
Exponential separation between positive and sign-changing solutions and its applications
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.
Kyudong Choi (UW Madison)
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We prove that k-th derivative of weak solutions is locally integrable in space-time for any real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of the initial data and on the domain of integration. Moreover, they are valid even for k >= 3 as long as we have a smooth solution. The proof uses a standard approximation of Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. Vasseur.
Yao Yao (UW Madison)
Confinement for nonlocal interaction equation with repulsive-attractive kernels
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.
Myoungjean Bae (POSTECH)
Transonic shocks for Euler-Poisson system and related problems
Abstract: Euler-Poisson system models various physical phenomena including the propagation of electrons in submicron semiconductor devices and plasmas, and the biological transport of ions for channel proteins. I will explain difference between Euler system and Euler-Poisson system and mathematical difficulties arising due to this difference. And, recent results about subsonic flow and transonic flow for Euler-Poisson system will be presented. This talk is based on collaboration with Ben Duan and Chunjing Xie.