# PDE Seminar 2009-2010

### Fridays 4:10pm, Stevenson Center 1307

Date: **Wednesday, September 30, 2009**

- Speaker:
**Roland Schnaubelt, University of Karlsruhe, Germany** - Title:
**Invariant manifolds and stability for the Stefan problem with surface tension** - Abstract: We study quasilinear parabolic systems with fully nonlinear dynamical boundary conditions, which arise e.g. after a transformation from problems with moving boundaries such as the Stefan problem with Gibbs-Thompson correction. We concentrate on the qualitative behavior near an equilibrium. Depending on the spectrum of the linearization, one obtains local stable, center and unstable invariant manifolds. We discuss their properties and the connection to the stability of the equilibrium. The proofs use maximal regularity results for inhomogeneous initial-boundary value problems, techniques from dynamical systems and semigroup theory.

Date: **Friday, October 9, 2009**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Asymptotic symmetry for positive solutions of parabolic problems** - Abstract: In this talk I will discuss qualitative properties of nonlinear parabolic problems. The basic question to be addressed is convergence of positive solutions to the space of symmetric functions. Sufficient conditions are formulated for such convergence in the case of a single equation and cooperative systems. The methods are based on the maximum principle, the Harnack inequality, and the method of moving hyperplanes.

Date: **Friday, October 16, 2009**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Asymptotic symmetry for positive solutions of parabolic problems, II** - Abstract: This talk is a continuation of the one given on October 9 and is devoted to the more technical aspects of the topic.

Date: **Friday, October 30, 2009**

- Speaker:
**Ugo Gianazza, Universita di Pavia, Italy** - Title:
**A new regularity approach for weak solutions of degenerate parabolic equations** - Abstract: In order to prove the Hoelder regularity of weak solutions to quasilinear degenerate parabolic equations, I use the same approach originally introduced in recent papers by DiBenedetto-Gianazza-Vespri to obtain Harnack inequalities for nonnegative solutions to these same equations. The new approach gives a more geometric and intuitive proof to the regularity and avoids covering and alternative arguments. This is joint work with M. Surnachev (U. of Swansea) and V. Vespri (U. of Florence).

Date: **Friday, November 6, 2009**

- Speaker:
**Zhian Wang, Vanderbilt University** - Title:
**Micro and macro models for chemotaxis** - Abstract: This talk is focused on two questions of chemotaxis modeling. One is how to establish the communications between microscopic and macroscopic chemotaxis models. The other is how information in the microscopic model is passed to the macroscopic model. For the first question, I use a novel approach to derive the macroscopic limits and express the microscopic quantities in terms of macroscopic quantities with the preservation of energy law. For the second question, I investigate the traveling waves of both microscopic and macroscopic models from which we see how traveling waves in the microscopic model are retained, lost or created during the transition from the microscopic to macroscopic models. Biological implications will be discussed along the talk.

Date: Friday, **November 13, 2009**

- Speaker:
**Leonardo Marazzi, Western Kentucky University** - Title:
**Scattering and inverse scattering on some classes of conformally compact manifolds** - Abstract: We study scattering theory on Asymptotically Hyperbolic (AH) manifolds and its generalizations to conformally compact manifolds and AH Einstein manifolds. Some examples of AH manifolds are the de Sitter-Schwarzschild model of the exterior of a black hole, which can be viewed as an AH manifold with two ends; and the Schwarzschild model of the exterior of a black hole, for which one of the two ends is a AH manifold, and the other end is an Asymptotically Euclidean manifold. Other examples of AH manifolds are given by quotients of the hyperbolic space by particular groups of motion. We also discuss some open problems in this area.

Date: **Friday, December 4, 2009**

- Speaker:
**Joanna Pressley, Vanderbilt University** - Title:
**Complementary responses to mean and variance modulations in the perfect integrate-and-fire model** - Abstract: In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.

Date: **Friday, February 5, 2010**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Liouville type theorems and a priori estimates for elliptic and parabolic problems, I** - Abstract: Using the method of moving hyperplanes we prove classical and new nonlinear Liouville type theorems. We also show how these theorems combined with scaling give rise to the a apriori estimates for elliptic and parabolic problems.

Date: **Friday, February 12, 2010**

- Speaker:
**Zhian Wang, Vanderbilt University** - Title:
**Modeling of cellular and population dynamics of chemotaxis** - Abstract: Chemotaxis is directed cell movement in response to external signal (or chemical). In this talk, we will discuss the different approaches of modeling cellular dynamics of chemotaxis and the derivation of the population model from the cellular models. Particularly we are interested in how to incorporate the individual cell properties into the macroscopic parameters.

Date: **Friday, February 19, 2010**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Liouville type theorems and a priori estimates for elliptic and parabolic problems, II** - Abstract: This talk is a continuation of the one given on February 5 and is devoted to more technical aspects of the topic.

Date:** Friday, February 26, 2010**

- Speaker:
**Xiaobing Feng, University of Tennessee, Knoxville** - Title:
**The vanishing moment method for fully nonlinear second order PDEs** - Abstract: In the past thirty years tremendous progresses have been made on the development of the viscosity solution theory for fully nonlinear 2nd order PDEs. However, in contrast with the success of the PDE theory, until very recently there has been essentially no progress on how to reliably compute these viscosity solutions. This lack of progress is due to the facts that (a) viscosity solutions often are only conditionally unique; (b) the notion of viscosity solutions is non-variational and nonconstructive, hence, it is extremely difficult (if it is possible) to mimic at the discrete level.

In this talk, I shall first review some recent advances (and attempts) in numerical methods for fully nonlinear 2nd order PDEs, in particular, the Monge-Ampere type PDEs. I shall then focus on discussing a newly developed moment solution theory, which is developed based on a (constructive) vanishing moment methodology, and the relationship between viscosity solutions and moment solutions. Recent developments in Galerkin type numerical methods (such as finite element methods, spectral methods, mixed methods) for fully nonlinear 2nd order PDEs based on the vanishing moment methodology will also be reviewed.

Finally, I shall present some numerical results for the Monge-Ampere equation, the prescribed Gauss curvature equation and the infinity-Laplace equation, and also discuss a few applications such as the semigeostrophic flow and the Monge-Kantorovich optimal mass transport which all give rise interesting and difficult fully nonlinear PDE problems. The results presented in this talk are based on a joint work with Michael Neilan of LSU.

Date: **Friday, March 5, 2010**

- Speaker:
**Gieri Simonett, Vanderbilt University** - Title:
**On the two-phase Navier-Stokes equations with surface tension** - Abstract: We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously, and we study the Rayleigh-Taylor instability.

Date: **Friday, March 12, 2010: Spring Break**

Date: **Friday, March 19, 2010**

- Speaker:
**Ivan Blank, Kansas State University** - Title:
**Sharp results for the regularity and stability of the free boundary in the obstacle problem** - Abstract: The problem of finding the smallest superharmonic function which lies above a given obstacle and which has prescribed boundary data is called the obstacle problem. After finding this minimizer, it is natural to study the regularity of the boundary of the set where the solution makes contact with the obstacle. In 1977 Caffarelli showed that for a very natural class of obstacles, when this (free) boundary is not smooth, it has to have a very specific geometry. This talk explores what happens to this result when we weaken Caffarelli’s hypotheses.

Date: **Friday, March 26, 2010**

- Speaker:
**Tilak Bhattacharya, Western Kentucky University** - Title:
**Properties of extreme values of infinity-harmonic functions and a local estimate** - Abstract: vu

Date: **Friday, April 9, 2010**

- Speaker:
**Jeremy LeCrone, Vanderbilt University** - Title:
**Continuous maximal regularity** - Abstract: In this talk I will discuss the notion of continuous maximal regularity (first introduced by DaPrato and Grisvard) and its implications for fully nonlinear parabolic equations.

Date: **Friday, April 16, 2010**

- Speaker:
**Wenzhang Huang, University of Alabama in Huntsville** - Title:
**The minimum wave speed of traveling waves for a Lotka-Volterra competition model** - Abstract: Consider a reaction-diffusion system that serves as a 2-species Lotka-Volterra competition model with each species having logistic growth in the absence of the other. Suppose that the corresponding reaction system has one unstable boundary equilibrium E_1 and one stable boundary equilibrium E_2. Then it is well known that there exists a positive number C_*, called the minimum wave speed, such that, for each c larger than or equal to C_*, the reaction-diffusion system has a positive traveling wave solution of wave speed c connecting E_1 and E_2, and the system has no nonnegative traveling wave with wave speed less than C_*. It has been shown that the minimum wave speed for this system is identical to another important quantity – the speed of the population spread towards to the stable equilibrium. Hence to find the minimum wave speed C_* not only is of the interest in mathematics but is of the importance in application. Although much research work has been done to give an estimate of C_* and some partial results have been obtained, the problem on finding an algebraic or analytic expression for the minimum wave speed remains unsolved in general. In this talk we will introduce a new, more efficient approach that enables us to determine precisely the minimum wave speed algebraically under conditions weaker than those given previously. We also show that the minimum wave speed in general cannot be determined by the linearization at the unstable equilibrium point. The conjecture on the precise minimum wave speed is also given.

Date: **Friday, April 23, 2010**

- Speaker:
**Martin Meyries, University of Karlsruhe, Germany** - Title:
**On parabolic systems with nonlinear dynamic boundary conditions** - Abstract: We present a theory for well-posedness of strong solutions for quasilinear parabolic systems with nonlinear dynamic (or Wentzell) boundary conditions, aiming at low initial regularity. The approach is based on a maximal regularity result for the inhomogeneous linearized problem in $L_p$-spaces with temporal weights. As an application we study the long-time behavior of solutions of reaction-diffusion systems with nonlinear boundary conditions of reactive-diffusive type.

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