(Jan 31) K. Trivisa: Hyperbolic conservation laws ---
Recent advances in the area of Hyperbolic Conservation Laws are discussed. Applications and models from Continuum Physics are described. The main focus of this talk is given to systems of conservation laws with large initial data. This is essentially an unexplored area of research. I present results on the existence and stability of solutions to hyperbolic systems of conservation laws with large BV data. The main tools in the analysis are the wave front tracking algorithm and the notion of an entropy functional.

(Feb 7) G. Dolzmann: Microstructure and nonconvex variational problems ---
Many properties of materials undergoing solid-solid phase transformations have been successfully predicted by minimizing the elastic energy in a continuum model. This approach leads typically to variational problems
with integrands that are not quasiconvex. Thus the direct method in the calculus of variations cannot be applied
to prove existence of minimizers. Minimizing sequences develop oscillations which are a mathematical
model for microstructures observed in experiments. In my talk I want to discuss some issues related to
nonconvex variational problems,  including relaxation, modeling relaxed energies, and the structure
of minimizers with finite surface energy.

(Feb 10) S-H. Yu: Riemann Problem of Viscous Conservation Laws ---
We consider the Riemann problem for systems of viscous conservation laws $u_t+ f(u)_x=u_{xx}$. It is the initial-value problem with simple jump discontinuity. The Riemann problem is the key to the understanding of wave phenomena for physical systems such as the Navier-Stokes equations in gas dynamics, particularly in the limit of zero dissipation.
The problem was solved by Riemann for the Euler equations in gas  dynamics and by Peter Lax for general system of hyperbolic conservation laws $u_t+f(u)_x)=0.$ The hyperbolic problem has the simple scaling $u(x,t)=\phi(x/t)$. For viscous conservation laws, there are much richer wave phenomena.  For instance, the local, intermediate and large-time behaviours of the solution are governed by different scalings. This is the consequence of the nonlinear coupling of hyperbolic waves in the presence of dissipations. We will describe the analysis of these wave behaviours.
 

(Feb 14) C. Doering: Laminar and turbulent energy dissipation in a shear layer with suction---
The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. For sufficiently large values of the suction rate, a steady laminar flow is absolutely stable at all shear rates. For sufficiently small but nonzero suction, however, the laminar flow is linearly unstable at high Reynolds numbers. We find that the rigorous upper bound on the energy dissipation rate---valid even for turbulent (and weak) solutions of the Navier-Stokes equations---scale precisely the same as the dissipation in the laminar solution in the zero viscosity limit. Both the laminar and any possible turbulent flows display a finite nonvanishing residual dissipation as the viscosity is decreased. This is a manifestation of so-called Kolmogorov-type scaling in which the energy dissipation rate becomes independent of the viscosity at high Reynolds numbers. This result establishes the sharpness of the upper bound's scaling in the vanishing viscosity limit for these boundary conditions. It also provides a mathematical illustration of the delicacy of corrections to high Reynolds number scaling (such as the logarithmic terms as appearing in the Prandtl-von Karman "law of the wall") to perturbations in the boundary conditions. This is joint work with Edward A. Spiegel (Columbia University) and Rodney A. Worthing (University of Michigan).

(Feb. 28) H. Koch: Unique continuation for ellipticequations with nonsmooth coefficients----
Let P be a differential operator and let V and W be measurable functions. We consider sufficiently regular functions u which satisfy |Pu| <= V |u| + W | gradient u |. We say the triple (P,V,W) has the strong unique continuation property (SUCP) if the following is true: u vanishes identically if it vanishes of infinite order at one point. Theorem (Koch and Tataru): Let n > 2 and let P be an elliptic partial differential operator of second order with Lipschitz continuous coefficients. The triple (P,V,W) has the SUCP if V in L^(n/2) and W in L^(n+epsilon). I will give a sketch of the proof and explain some of its implications and the relation to questions occuring in the context of nonlinear dispersive equations.

(March 9) Zhouping Xin: Global Weak Solutions to A Shallow Water Equation----
In this talk, I will present a result on the existence of global in time weak solutions to the Cauchy problem for an one-dimensional shallow water equation, which is formally integrable and can be obtained by approximating directly the Hamiltonian for the Euler's equation in the shallow water regime. This system has the interesting property that is formally integrable but admits wave breaking. The main goal is to obtain $H^1$-bounded solutions. Our solutions are obtained as limits of viscous approximations. The key elements in our analysis are some new a-priori one-sided super-norm and local space-time estimates for the first order derivatives. The strong convergence in $H^1$ is obtained by estimating the associated Young measures.

(March 20 ) Guido Schneider: Stability of Modulated Fronts for the Swift-Hohenberg Equation---
We consider front solutions of the Swift-Hohenberg equation $\partial _t u=-(1+\partial _x^2)^2 u +\epsilon ^2 u -u^3$. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using renormalization techniques and a decomposition into Bloch waves, we show the non-linear stability of these solutions. It turns out that this problem is closely related to the question of stability of the trivial solution for the model problem $\partial _t u(x,t) = \partial _x^2 u (x,t)+(1+\tanh(x-ct))u(x,t)+u(x,t)^p$ with $p>3$. In particular, we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front.

(March 27) :C. Schober: The higher order nonlinear Schrödinger equation and the evolution of deep water waves---
In this talk we experimentally and theoretically examine the long time evolution of modulated periodic 1-D Stokes waves which are described, to leading order, by the nonlinear Schr\"{o}dinger (NLS) equation. The laboratory and numerical experiments indicate that under suitable conditions modulated periodic wavetrains evolve chaotically. A Floquet spectral decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave states to left and right going modulated traveling waves. Numerical experiments using a higher order nonlinear Schr\"{o}dinger equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary conditions the long-time evolution of modulated wavetrains is chaotic. Further, the numerical experiments indicate that the macroscopic features of the evolution can be modeled by the HONLS equation. Ultimately, these laboratory experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.

( April 3) Yu Yuan: A priori estimates for two classes of fully nonlinear equations without convexity---
(I) We derive an a priori estimate for the fully nonlinear elliptic equations with convex level sets. We do not need any convexity assumption for the proof of two dimensional case, as the classical result indicates. This is a joint work with L. A. Caffarelli. (II) We also derive an a priori estimate in dimension three for the special Lagrangian equations, which fail both the usual convexity condition and the assumption in (I).

(April 10) M. Dauge: A new method for using nodal finite elements in Maxwell computations for domains with non-convex corners---
It is now well known that using nodal finite elements in a conformal variational formulation involving the regularized bilinear form rot-rot + div-div with essential boundary conditions leads to definitely wrong results in the presence of non-convex corners. We present a new way of regularizing the rot-rot operator by a temperate divergence term, with the help of a weight. We will provide theoretical arguments and numerical results, based on a joint work with Martin Costabel and on a code developped by Daniel Martin.

(April 20) N. Masmoudi: From Boltzmann equations to Fluid Mechanics equations ---
We consider here the problem of deriving rigorously, globally in time and for general initial conditions, fluid mechanics equations such as Navier-Stokes, Euler or Stokes equations from the Boltzmann's equations. Our results may be viewed as extensions of the series of works by C. Bardos, F. Golse and D. Levermore. The methods used here are very much related to those used for the study of low Much number limits (i.e the convergence of solutions of compressible, isentropic, Navier-Stokes equations to those of incompressible equations).

(April 24) I. Rodnianski: Smoothing effects for the Schrödinger evolution ---
We study the spatial regularity of the solutions of the linear time-dependent Schr\"odinger equation. The first case considered is the motion of a quantum particle in the electric field of a subquadratic potential. We describe the connection between the decay of the initial data at infinity and the consequtive improvement of regularity. We then consider the periodic Schr\"odinger equation and establish that the "smoothness" of the propagator at time $\,t\,$ depends upon the diophantine properties of $\,t$. The formation of "quantum fractals" is also discussed.

(May 1) C. Carstensen: Finite Element Methods for Nonconvex Minimisation Problems and an Application in Material Science----
The rapidly developing field of the mathematical modeling of microstructure has important applications in material science (advanced materials), micromagnetism, homogenization and optimization. Typically, the continuous problem (P) lacks classical solutions. There exist minimizing sequences in (P) that have a weak limit, but non-(quasi-)convexity implies, in general, that the weak limit u is {\em not} a solution of the problem (P). In experiments, we observe oscillations of strains which form a macroscopic or averaged quantity u. The efficient numerical simulation on the macroscopic level aims to compute the weak limit u as a solution of a related {\em Relaxed Problem} (RP) while the microscopic mechanism can be detected by a direct finite element minimization of (P) or, more sophisticated, by a generalised formulation (GP).
The presentation will focus on adaptive strategies for relaxed problems such as the double-well problem in one-dimension (Young's example), in higher dimensions, or in linearised elasticity and on related topics in micromagnetism and homogenization problems.

(May 15) T.Wanner: Transient Pattern Formation in the Cahn-Hilliard Model----
The Cahn-Hilliard equation $u_t = -\Delta (\epsilon^2 \Delta u + f(u))$ was introduced as a model for several physical phenomena occurring in binary metal alloys. One particularly intriguing phenomenon is spinodal decomposition: If a homogeneous high-temperature mixture of two metallic components is rapidly quenched to a certain lower temperature, then a sudden phase separation sets in. The mixture quickly becomes inhomogeneous and forms a complicated, fine-grained structure, more or less alternating between the two alloy components. In this talk I present recent mathematical results explaining spinodal decomposition for the Cahn-Hilliard equation. This includes both a description of the main characteristics of the generated patterns, as well as an explanation of the mechanism underlying the phase separation. I will also address the case of multi-component alloys, which is modeled by a system of Cahn-Hilliard equations, as well as the stochastically perturbed Cahn-Hilliard-Cook equation.