Seminars and Colloquia by Series

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.

Construction of quasi-periodic attractors for systems with strong damping

Series
PDE Seminar
Time
Tuesday, September 3, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Renato Calleja – UNAM, Mexico
I will present a method for constructing periodic or quasi-periodic solutions for forced strongly dissipative systems. Our method applies to the varactor equation in electronic engineering and to the forced non-linear wave equation with a strong damping term proportional to the wave velocity. The strong damping leads to very few small divisors which allows to prove the existence by using a fixed point contraction theorem. The method also leads to efficient numerics. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.

Reductions of fluid and kinetic equations using Dirac'stheory of constrained Hamiltonian systems

Series
PDE Seminar
Time
Tuesday, August 20, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
006
Speaker
Cristel Chandre – Center for Theoretical Physics, Univ. Aix-Marseille
Dirac'stheory of constrained Hamiltonian systems allows for reductions of the dynamics in a Hamiltonian framework. Starting from an appropriate set of constraints on the dynamics, Dirac'stheory provides a bracket for the reduced dynamics. After a brief introduction of Dirac'stheory, I will illustrate the approach on ideal magnetohydrodynamics and Vlasov-Maxwell equations. Finally I will discuss the conditions under which the Dirac bracket can be constructed and is a Poisson bracket.

On well-posedness for a class of first order Hamilton-Jacobi equation in metric spaces

Series
PDE Seminar
Time
Tuesday, April 23, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jin Feng – University of Kansas
Using metric derivative and local Lipschitz constant, we define action integral and Hamiltonian operator for a class of optimal control problem on curves in metric spaces. Main requirement on the space is a geodesic property (or more generally, length space property). Examples of such space includes space of probability measures in R^d, general Banach spaces, among others. A well-posedness theory is developed for first order Hamilton-Jacobi equation in this context. The main motivation for considering the above problem comes from variational formulation of compressible Euler type equations. Value function of the variation problem is described through a Hamilton-Jacobi equation in space of probability measures. Through the use of geometric tangent cone and other properties of mass transportation theory, we illustrate how the current approach uniquely describes the problem (and also why previous approaches missed). This is joint work with Luigi Ambrosio at Scuola Normale Superiore di Pisa.

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