(Jan. 10) J. Smoller Title: Nonlinear Dynamical Stability of Rotating White Dwarfs and Supermassive Stars ---
We prove nonlinear stability and existence theorems for rotating star solutions which are axi-symmetric steady-state solutions of the compressible isentropic Euler-Poisson equations in 3 spatial dimensions. We apply our results to rotating and non-rotating white dwarf, and rotating supermassive (extreme relativistic) stars, stars which are in convective equilibrium and have uniform chemical composition. (This is joint work with Tao Luo.)

(Jan. 17) O. Calin Title: Finding Heat Kernels using Geometric Mechanics ---
Geometric Mechanics has useful and important applications to PDEs and quantum mechanics. We shall present a geometric method which deals with the construction of heat kernels for Hermite-type operators, Kolmogorov operators as well as operators with linear, quadratic and inverse square potential in one and several dimensions.

(Jan. 24) M. Guha Title: Front Propagation in a Noisy, Nonsmooth, Excitable Medium ---
We consider the impact of noise on the stability and propagation of fronts in an excitable media with a piece-wise smooth, discontinuous ignition process. In a neighborhood of the ignition threshold the system interacts strongly with noise, the front can loose monotonicity, resulting in multiple crossings of the ignition threshold. We adapt the renormalization group methods developed for coherent structure interaction, a key step being to determine pairs of function spaces for which the ignition function is Frechet differentiable, but for which the associated semi-group, S(t), is integrable at t=0.
We parameterize a neighborhood of the front solution through a dynamic front position and a co-dimension one remainder. The front evolution and the asymptotic decay of the remainder are on the same time scale, the RG approach shows that the remainder becomes asymptotically small, in terms of the noise strength and regularity, and the front propagation is driven by a competition between the ignition process and the noise.

(Jan. 31) P. Smereka Title: Long Time Behavior of a Modified Becker-D\"oring System ---
A modification, based on asymptotic behavior, of the Becker-D\"oring system is introduced in which the concentration of monomers is slaved to the concentrations of the other clusters. This modified system has the same continuum limit as usual Becker-D\"oring system. For one member of these it is proved, for compact initial data, that all solutions will converge to a unique self-similar solution as time tends to infinity.

(Feb.21) R. Temam Title: The three-dimensional linearized Primitive Equations of the atmosphere and the oceans ---
Whereas the Primitive Equations with viscosity bear a certain similarity with the incompressible Navier Stokes equations, it is well-known that, in the inviscid case, the Primitive Equations do not ressemble the incompressible Euler equations. In this lecture we will explain the difficulties related to the inviscid Primitive Equations and describe results of existence and uniqueness of solutions in the linearized case (after an article with A. Rousseau and J. Tribbia)

(Mar.6) C. Doering Title: Statistically Stationary Stirring with Steady Sources and Sinks ---
We discuss applications, models, analysis and simulation of mixing of a passive scalar field sustained by temporally steady but spatially inhomogeneous sources. We consider a broad class of statistically stationary incompressible flows and measure the mixing in terms of scalar field variance supression in the presence of the stirring relative to that in the presence of molecular diffusion alone. Notions of "eddy diffusion" or "effective diffusion" of flows are discussed.

(Mar.13) Tao Luo Title: Transonic Shock Solutions for a System of Euler-Poisson Equations ---
A boundary value problem for a system of Euler-Poisson equations modeling semiconductor devices or plasma is considered. The boundary conditions are supersonic on supersonic inflow and subsonic outflow. The purpose is to elucidate the role played by the electric filed to the structure of solutions with transonic shocks. The existence, non-existence, uniqueness and non-uniqueness of solutions with transonic shocks are obtained according to the different cases of boundary data and physical interval length. Detailed structures of solutions are given. Shock locations are determined by the boundary data. Different phenomena are show for the different situations when the density of fixed, positively charged background ions is in supersonic and subsonic regimes. This is a joint work with Zhouping Xin.

(Mar.20) Fei-Ran Tian Title: Initial Value Problem of the Whitham equations for the Camassa-Holm equation ---
Like the KdV equation, the Camassa-Holm equation with small dispersion can be viewed as a dispersive approximation to the inviscid Burgers equation. Although the zero dispersion limit for the Camassa-Holm equation has not been established, the modulation equations (i.e., Whitham equation) have been derived. Unlike the KdV case, the Whitham equations are non-strictly hyperbolic. In this talk, we will discuss the initial value problem related to these Whitham equations.

(Mar.27) Sijue Wu Title: Almost global wellposedness of the 2-D full water wave equation ---
We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of type $\epsilon\Phi$, where $T$ depends only on $\Phi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/{\epsilon}}]$. This is achieved by better understandings of the nature of the nonlinearity of the water wave equation.

(April 3) David Hoff Title: Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow ---
Abstract: We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories.

(April 10) Li Chen Title: Analysis on Some Parabolic Systems (strongly coupled or higher order) ---
Abstract: In this talk, I will give a brief outline of the works we have done on some stronly coupled parabolic systems and a forth order parabolic equation, on which such classical technics as comparison principle for parabolic quations could not work. The models are from diffusion systems in semiconductor simulation and population models in biomathematical model. Our idea are mainly based on the exponential transfromation and entropy inequalities. We will give the global existences and large time behavior of the weak solutions.