Tuesday, October 23, 2012 - 15:05 , Location: Skiles 006 , Alexander Kiselev , Department of Mathematics, University of Wisconsin, Madison , Organizer: Zhiwu Lin
Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic (SQG) equations. Many questions about regularity and properties of solutions of these equations remain open. I will discuss the recently introduced idea of nonlocal maximum principle, which helped prove global regularity of solutions to the critical SQG equation. I will describe some further recent developments on regularity and blowup of solutions to active scalar equations.
Tuesday, October 9, 2012 - 15:05 , Location: Skiles 006 , Roman Shvydkoy , University of Illinois at Chicago , Organizer: Zhiwu Lin
The existence of self-similar blow-up for the viscous incompressible fluids was a classical question settled in the seminal of works of Necas, et al and Tsai in the 90'. The corresponding scenario for the inviscid Euler equations has not received as much attention, yet it appears in many numerical simulations, for example those based on vortex filament models of Kida's high symmetry flows. The case of a homogeneous self-similar profile is especially interesting due to its relevance to other theoretical questions such the Onsager conjecture or existence of Landau type solutions. In this talk we give an account of recent studies demonstrating that a self-similar blow-up is unsustainable the Euler system under various weak decay assumptions on the profile. We will also talk about general energetics of the Euler system that, in part, is responsible for such exclusion results.
Monday, October 8, 2012 - 16:05 , Location: Skiles 006 , Christoph Walker , University of Hannover, Germany , Organizer: Zhiwu Lin
The talk focuses on positive equilibrium (i.e. time-independent)solutionsto mathematical models for the dynamics of populations structured by ageand spatial position. This leads to the study of quasilinear parabolicequations with nonlocal and possibly nonlinear initial conditions. Weshallsee in an abstract functional analytic framework how bifurcationtechniquesmay be combined with optimal parabolic regularity theory to establishtheexistence of positive solutions. As an application of these results wegivea description of the geometry of coexistence states in a two-parameterpredator-prey model.
An Invitation to the Millennium Prize Problem for the Navier-Stokes Equation and its Probabilistic CounterpartTuesday, October 2, 2012 - 15:05 , Location: Skiles 006 , S. S. Sritharan , Naval Postgraduate School, Monterey, California , Organizer: Zhiwu Lin
In this talk we will give a very elementary explanation of issues associated with the unique global solvability of three dimensional Navier-Stokes equation and then discuss various modifications of the classical system for which the unique solvability is resolved. We then discuss some of the fascinating issues associated with the stochastic Navier-Stokes equations such as Gaussian & Levy Noise, large deviations and invariant measures.
Tuesday, September 25, 2012 - 15:05 , Location: Skiles 006 , Meijun Zhu , University of Oklahoma , Organizer: Zhiwu Lin
We shall describe our recent work on the extension of sharp Hardy-Littlewood-Sobolev inequality, including the reversed HLS inequality with negative exponents. The background and motivation will be given. The related integral curvature equations may be discussed if time permits.
Tuesday, September 18, 2012 - 15:05 , Location: Skiles 006 , Bin Cheng , Arizona State University , Organizer: Zhiwu Lin
Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a PDE-theoretical framework for studying time-averages of dynamical systems that evolve in both fast and slow scales. Patterns arise upon time-averaging, which in turn affects long term dynamics via nonlinear coupling. We apply this framework to geophysical fluid dynamics in spherical and bounded domains subject to strong Coriolis force and/or Lorentz force.
Tuesday, September 4, 2012 - 15:05 , Location: Skiles 006 , Zhaoyang Yin , Sun Yat-sen University, China , Organizer: Zhiwu Lin
In this talk, we consider the Cauchy problem of a modified two-component Camassa-Holm shallow water system. We first establish local well-possedness of the Cauchy problem of the system. Then we present several blow-up results of strong solutions to the system. Moreover, we show the existence of global weak solutions to the system. Finally, we address global conservative solutions to the system. This talk is based on several joint works with C. Guan, K. H. Karlsen, K. Yan and W. Tan.
Tuesday, May 1, 2012 - 10:05 , Location: Skiles 005 , Clement Mouhot , University of Cambridge , Organizer: Andrzej Swiech
Mark Kac proposed in 1956 a program for deriving the spatially homogeneous Boltzmann equation from a many-particle jump collision process. The goal was to justify in this context the molecular chaos, as well as the H-theorem on the relaxation to equilibrium. We give answers to several questions of Kac concerning the connexion between dissipativity of the many-particle process and the limit equation; we prove relaxation rates independent of the number of particles as well as the propagation of entropic chaos. This crucially relies on a new method for obtaining quantitative uniform in time estimates of propagation of chaos. This is a joint work with S. Mischler.
Tuesday, April 24, 2012 - 15:05 , Location: Skiles 005 , Xiaoyi Zhang , University of Iowa , Organizer: Andrzej Swiech
In this talk, we consider 3d defocusing energy critical NLS on the exterior domain of a convex obstacle with Dirichlet boundary condition. We show that all solutions with finite energy exist globally and scatter.
Thursday, April 12, 2012 - 15:05 , Location: Skiles 005 , Guillaume Carlier , Universite de Paris IX (Paris-Dauphine) , Organizer: Andrzej Swiech
In this talk, I will describe several models arising in congested transport problems. I will first describe static models which lead to some highly degenerate elliptic PDEs. In the second part of the talk, I will address dynamic models which can be seen as a generalization of the Benamou-Brenier formulation of the quadratic optimal transport problem and will discuss the existence and regularity of the adjoint state. The talk will be based on several joint works with Lorenzo Brasco, Pierre Cardaliaguet, Bruno Nazaret and Filippo Santambrogio.