Reactive Processes in Inhomogeneous Media

Speaker: 

Andrej Zlatos

Institution: 

University of Wisconsin at Madison

Time: 

Thursday, March 31, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

We study fine details of spreading of reactive processes in multidimensional
inhomogeneous media. In the real world, one often observes a transition from one equilibrium (such as unburned areas in forest fires) to another (burned areas)to happen over short spatial as well as temporal distances. We demonstrate that this phenomenon also occurs in one of the simplest models of reactive processes, reaction-diffusion equations with ignition reaction functions, under very general hypotheses.

Specifically, in up to three spatial dimensions, the width (both in space and time) of the zone where the reaction occurs turns out to remain uniformly bounded in time for fairly general classes of initial data. This bound even becomes independent of the initial data and of the reaction function after an initial time interval. Such results have recently been obtained in one dimension, in which one can even completely characterize the long term dynamics of general solutions to the equation, but are new in dimensions two and three. An indication of the added difficulties is the fact that three dimensions turns out to indeed be the borderline case, as the bounded-width result is in fact false for general inhomogeneous media in four and more dimensions.

A Degenerate Isoperimetric Problem and its Relation to Traveling Waves for a Bi-Stable Hamiltonian System

Speaker: 

Peter Sternberg

Institution: 

Indiana University

Time: 

Tuesday, May 31, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

I will discuss a simple-looking isoperimetric problem for curves in the plane where length is measured with respect to a degenerate metric. One motivation for the study is that geodesics for this problem, appropriately parametrized, lead to traveling waves associated with a Hamiltonian system based on a bi-stable potential. This is joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok.

Positivity of the complex Laplacian and its applications

Speaker: 

Siqi Fu

Institution: 

Rutgers University

Time: 

Thursday, April 21, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

In this expository talk we will discuss aspects of spectral theory of the complex Laplacian, revolving around
the notion of positivity.  We will discuss geometric/potential theoretic characterizations
for positivity of the complex Neumann Laplacian and explain some applications of the theory in complex geometry.

On Thomas-Fermi Theory and Extensions

Speaker: 

Gisele Goldstein

Institution: 

University of Memphis

Time: 

Thursday, April 28, 2016 - 3:00pm

Location: 

RH 440R

Of concern to quantum chemists and solid state physicists is the approximate numerical computation of the ground state wave function, and the ground state energy and density for molecular and other quantum mechanical systems. Since the number of molecules in bulk matter is of the order of 10e26 , direct computation is too cumbersome or impossible in many situations. In 1927, L. Thomas and E. Fermi proposed replacing the ground state wave function by the ground state density, which is a function of only three variables. Independently, each found an approximate expansion for the energy associated with a density. (The wave function uniquely determines the density, but not conversely.)

A computationally better approximate expansion was found in the 1960’s by W. Kohn and his collaborators; for this work Kohn got the Nobel Prize in Chemistry in 1998. A successful attempt to put Thomas-Fermi theory into a rigorous mathematical framework was begun in the 1970’s by E. Lieb and B. Simon and was continued and expanded by Ph. Benilan, H. Brezis and others. Very little rigorous mathematics supporting Kohn density functional theory is known. In this talk I will present a survey of rigorous Thomas-Fermi theory, including recent developments and open problems (in the calculus of variations and semilinear elliptic systems).

On nematic liquid crystal flows in dimensions two and three

Speaker: 

Changyou Wang

Institution: 

Purdue University

Time: 

Tuesday, February 2, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

In this talk, I will discuss a simplified Ericksen-Leslie system modeling
the hydrodynamics of nematic liquid crystals, that is coupling between Navier-Stokes equations and harmonic map heat flows. I will describe some existence results of global weak solutions in dimensions two and three, and a finite time singularity result in dimension three. This is based on some joint works with Tao Huang, Junyu Lin, Fanghua Lin, and Chun Lin.

Line Defects in a Modified Ericksen Model of Nematic Liquid Crystals

Speaker: 

Robert M Hardt

Institution: 

Rice University

Time: 

Monday, October 12, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit orientation vectorfield on a region of R^3  but also a scalar order parmeter quantify- ing the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2. He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular de- fect set s^{-1}(0). The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, led to regularity away from isolated points, which unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions using RP2. In recent ongo- ing work with FH.Lin and O. Alper, we see that the singular set with this model necessarily consists of Holder continuous curves. We will also survey some of the many more elaborate liquid crystal PDE’s involving a general director functional, the full Q tensor model, and possible coupling with fluid velocity. 

Coating flow of viscous Newtonian liquids on a rotating cylinder

Speaker: 

Marina Chugunova

Institution: 

Claremont Graduate University

Time: 

Thursday, October 22, 2015 - 3:00pm

Location: 

RH 440R

In this talk I discuss the different types of models which originate from a lubrication approximation of viscous coating flow dynamics on the outer surface of a rotating cylinder, that is in the presence of a gravitational field. Analytical and numerical results related to existence, uniqueness and stability of solutions will be presented.

Homogenization of Oscillating Boundary Conditions

Speaker: 

William Feldman

Institution: 

UCLA

Time: 

Thursday, May 14, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

I will discuss the homogenization of periodic oscillating Dirichlet
boundary problems in general domains for second order uniformly elliptic
equations. These problems are connected with the study of boundary layers
in fluid mechanics and with the study of higher order asymptotic expansions
in interior homogenization theory. The talk will be aimed at a general
audience. I will explain some recent progress about the continuity
properties of the homogenized problem which displays a sharp contrast
between the case of linear and nonlinear interior equations. This is based
on joint work with Inwon Kim.

Global in time Gevrey regularity solution for a class of nonlinear gradient flows

Speaker: 

Cheng Wang

Institution: 

University of Massachusetts, Dartmouth

Time: 

Tuesday, March 17, 2015 - 3:00pm

Location: 

RH 306

The existence and uniqueness of Gevrey regularity solution for a class of nonlinear
bistable gradient flows, with the energy decomposed into purely convex and concave parts,
such as epitaxial thin film growth and square phase field crystal models, are discussed in this talk.
The polynomial pattern of the nonlinear terms in the chemical potential enables one to derive a
local in time solution with Gevrey regularity, with the existence time interval length dependent
on certain functional norms of the initial data. Moreover, a detailed Sobolev estimate for the gradient
equations results in a uniform in time bound, which in turn establishes a global in
time solution with Gevrey regularity. An extension to a system of gradient flows,
such as the three-component Cahn-Hilliard equations, is also addressed in this talk.

Pages

Subscribe to RSS - Nonlinear PDEs