# PDE Seminar 2008-2009

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

Date: **Friday, 5 September 2008**

- Speaker:
**Misha Perepelitsa, Vanderbilt University** - Title:
**Dynamics of a density discontinuity in compressible viscous flows.** - Abstract: We will discuss the evolution of the interface of a density discontinuity in the model of Navier-Stokes equations for compressible flows. All time existence of weak, near equilibrium solutions will be established, when the interface is in non-singular contact with the boundary of the flow domain. A singular contact problem will be considered on a model system of equations for which we show instantaneous change of some geometrical properties of the interface.

Date: **Friday, 12 September 2008**

- Speaker:
**Glenn Webb, Vanderbilt University** - Title:
**Analysis of a model for transfer phenomena in biological populations** - Abstract: The problem of transfer in a population structured by a continuous quantity is analyzed. The transfer of the quantity occurs between individuals according to specified rules. The simple model is an ordinary differential equation in the Banach space of integrable functions, of Boltzmann type with kernel corresponding to a transfer process. It is proved that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. The simple model is generalized by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that the transfer rate is sufficiently small. An application is made to a model of proliferating cell populations with individual cells exchangimg the surface protein P-glycoprotein, which plays an important role in acquired multidrug resistance against cancer chemotherapy.

Date:** Friday, 19 September 2008**

- Speaker:
**Gieri Simonett, Vanderbilt University** - Title:
**On normal stability for nonlinear parabolic equations** - Abstract: We study convergence of solutions for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-dicrete, but forms a C^1-manifold which is normally stable. Our approach uses tools from the theory of maximal regularity in an essential way.

Date: **Friday, 26 September 2008**

- Speaker:
**Joanna Pressley, Vanderbilt University** - Title:
**Response dynamics of integrate-and-fire neuron models** - Abstract: The brain is a complex network composed of more than 100 billion neurons, each making thousands of connections with other neurons. Neurons use pulse-like electrical signals called action potentials or “spikes” to encode information. The rate of spikes, or the firing rate, is believed to contain a majority of the pertinent information transferred in a spike train.
One of the fundamental problems in neuroscience is characterizing the transfer function that converts noisy synaptic inputs into output firing rates. Using Fokker-Planck formalism, we describe the firing rate response dynamics of integrate-and-fire neuron models. Model dynamics are complex, with significant nonlinearities and heightened responses at certain frequencies of input. Elucidating the response properties of simplified stochastic models is a crucial step toward understanding how the brain performs reliable, temporally precise computations using circuits composed of noisy neurons.

Date: **Friday, 17 October 2008**

- Speaker:
**Pasquale Candito, Mediterranea University of Reggio Calabria, Italy** - Title:
**Critical point theory and its applications** - Abstract: The aim of the talk is to give some recent critical point results for nondifferentiable functionals and its applications to elliptic partial differential equations with discontinuous nonlinearities as well as to discrete boundary value problems.

Date: **Friday, 31 October 2008**

- Speaker:
**Mike Frazier, University of Tennessee, Knoxville** - Title:
**Estimates for Green’s functions of Schrodinger operators** - Abstract: We consider the inhomogeneous, time-independent Schrodinger equation. Under certain conditions on the potential, we obtain global lower and upper exponential estimates for the Green’s function of the Schrodinger operator in terms of the first and second iterates of the Green’s function for the Laplacian. The estimates hold on the whole space and for a very general class of domains.
The results for Schrodinger operators are a consequence of a more general result. If T is a bounded linear operator on L^2 (\mu) with norm less than one, then I-T has an inverse given by a Neumann series. Suppose T is represented by integration against a symmetric kernel K(x,y). Under the condition that the reciprocal of K is a quasimetric, we obtain global exponential bounds (both lower and upper, but with different constants) for the kernel of the inverse of I-T.

Our methods also apply to operators with fractional potential replacing the Laplacian. These operators relate to alpha-stable Levy processes in the same way that the Laplacian relates to Brownian motion. (Joint work with Fedor Nazarov and Igor Verbitsky)

Date: **Friday, 7 November 2008**

- Speaker:
**Michael O’Leary, Towson University** - Title:
**A diffusion model in population genetics with mutation and dynamic fitness** - Abstract: We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection, drift and mutation. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system provided the mutation rate is sufficiently small. We also analyze the long term limit of the genetic variance in the system.

Date: **Thursday, 12 March 2009, Colloquium talk (4:10pm, SC 5211)**

- Speaker:
**Gui-Qiang Chen, Northwestern University** - Title:
**Shock Reflection-Diffraction Phenomena, von Neumann’s Conjecture, and Nonlinear PDEs of Mixed Type** - Abstract: We will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the shock reflection-diffraction problems can be formulated into free boundary problems for nonlinear PDEs of mixed-composite hyperbolic-elliptic type and present the von Neumann’s sonic conjecture. The problems involve two types of transonic flow: One is a continuous transition through a pseudo-sonic circle, and the other is a jump transition through the transonic shock as a free boundary. Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including some recent results on the existence, stability, and regularity of global solutions of shock reflection-diffraction by wedges. This talk will be based mainly on joint work with M. Feldman.

Date: **Thursday, 9 April 2009, Colloquium talk (4:10pm, SC 5211)**

- Speaker:
**James Serrin, University of Minnesota** - Title:
**Entire solutions of completely coercive quasilinear elliptic equations** - Abstract: A famous theorem of Sergei Bernstein says that every entire solution u = u(x), x in R², of the minimal surface equation, div{Du(1 + Du²)^(-1/2)} = 0, is a linear function; no conditions whatsoever being placed on the behavior of the solution u. This result however fails to be true in higher dimensions, in fact if x in R^n, with n > 7, there exist entire non-constant solutions (Bombiere, DiGiorgi and Miranda). Our purpose is to consider other quasilinear elliptic equations which do have the following Bernstein–Liouville property, namely that u is constant for any entire solution u, but where no restrictions are placed on the dimension n and no conditions are assumed on the behavior of the solution. Note that the Laplace equation Delta u = 0 does not qualify: while bounded solutions, or even solutions which are bounded either above or below, are constant (the Liouville theorem), one easily finds entire non-constant solutions (of course bounded neither above nor below): e.g., even in two dimensions the solution x² – y² is a case in point.

Date: **Friday, April 10, 2009**

- Speaker:
**James Serrin** - Title:
**Entire solutions of completely coercive quasilinear elliptic equations, II** - Abstract: In this talk, proofs and technical details will be given.

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