Seminars and Colloquia by Series

HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH DISSIPATION

Series
PDE Seminar
Time
Tuesday, April 15, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Constantine Dafermos – Brown University
ABSTRACT: The lecture will outline a research program which aims at establishing the existence and long time behavior of BV solutions for hyperbolic systems of balance laws, in one space dimension, with partially dissipative source, manifesting relaxation. Systems with such structure are ubiquitous in classical physics.

Mixed type problems in transonic flow and isometric embedding

Series
PDE Seminar
Time
Tuesday, March 11, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dehua Wang – University of Pittsburgh
Some mixed-type PDE problems for transonic flow and isometric embedding will be discussed. Recent results on the solutions to the hyperbolic-elliptic mixed-type equations and related systems of PDEs will be presented.

GLOBAL SMOOTH SOLUTIONS IN R^3 TO SHORT WAVE-LONG WAVE INTERACTIONS SYSTEMS FOR VISCOUS COMPRESSIBLE FLUIDS

Series
PDE Seminar
Time
Tuesday, March 4, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hermano Frid – IMPA, Brazil
The short wave-long wave interactions for viscous compressibleheat conductive fluids is modeled, following Dias & Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schrodingerequation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. This is a joint work with Ronghua Panand Weizhe Zhang.

Some New Comparison Results in Balls and Shells

Series
PDE Seminar
Time
Tuesday, February 18, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeffrey Langford – Bucknell University
In a comparison theorem, one compares the solution of a given PDE to a solution of a second PDE where the data are "rearranged." In this talk, we begin by discussing some of the classical comparison results, starting with Talenti's Theorem. We then discuss Neumann comparison results, including a conjecture of Kawohl, and end with some new results in balls and shells involving cap symmetrization.

$L^2$-geometry of diffeomorphism groups and the equations of hydrodynamics

Series
PDE Seminar
Time
Tuesday, January 28, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gerard Misiolek – University of Notre Dame
In 1966 V. Arnold observed that solutions to the Euler equations of incompressible fluids can be viewed as geodesics of the kinetic energy metric on the group of volume-preserving diffeomorphisms. This introduced Riemannian geometric methods into the study of ideal fluids. I will first review this approach and then describe results on the structure of singularities of the associated exponential map and (time premitting) related recent developments.

Blowup criterion for the strong solutions to 3D incompressible Navier-Stokes equations in BMO^{-s} spaces

Series
PDE Seminar
Time
Tuesday, January 21, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jianli Liu – Shanghai Unversity
This talk gives a blowup criteria to the incompressible Navier-Stokes equations in BMO^{-s} on the whole space R^3, which implies the well-known BKM criteria and Serrin criteria. Using the result, we can get the norm of |u(t)|_{\dot{H}^{\frac{1}{2}}} is decreasing function. Our result can obtained by the compensated compactness and Hardy space result of [6] as well as [7].

Self-Diffusion and Cross-Diffusion Equations: $W^{1,p}$-Estimates and Global Existence of Smooth Solutions

Series
PDE Seminar
Time
Tuesday, December 3, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tuoc V. Phan – University of Tennessee, Knoxville
We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension.We obtain this result by deriving global $W^{1,p}$-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation techniquetogether with a new two-parameter scaling argument.The talk is based on my joint work with Luan Hoang (Texas Tech University) and Truyen Nguyen (University of Akron)

Ricci curvature for finite Markov chains

Series
PDE Seminar
Time
Tuesday, November 19, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matthias Erbar – University of Bonn
In this talk I will present a new notion of Ricci curvature that applies to finite Markov chains and weighted graphs. It is defined using tools from optimal transport in terms of convexity properties of the Boltzmann entropy functional on the space of probability measures over the graph. I will also discuss consequences of lower curvature bounds in terms of functional inequalities. E.g. we will see that a positive lower bound implies a modified logarithmic Sobolev inequality. This is joint work with Jan Maas.

Recent progress for large data solutions on compressible Euler equations

Series
PDE Seminar
Time
Tuesday, October 1, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geng Chen – Georgia Tech –
It is well known that solutions of compressible Euler equations in general form discontinuities (shock waves) in finite time even when the initial data is $C^\infty$ smooth. The lack of regularity makes the system hard to resolve. When the initial data have large amplitude, the well-posedness of the full Euler equations is still wide open even in one space dimenssion. In this talk, we discuss some recent progress on large data solutions for the compressible Euler equations in one space dimension. The talk includes joint works with Alberto Bressan, Helge Kristian Jenssen, Robin Young and Qingtian Zhang.

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