(Sept. 11) V. Elling Title: Counterexamples to the sonic and detachment criteria ---
When waves meet an oblique wall, they usually undergo regular reflection. But for shock waves there are two kinds of reflection, weak and strong. Moreover around 1875 Ernst Mach discovered another type of reflection, now named after him. Ever since, researchers have tried to find simple conditions predicting which type of reflection appears in particular circumstances. The most important candidates are the detachment, sonic, and von Neumann criteria.
We prove counterexamples showing that both the sonic and detachment criterion are not universally correct. In particular there is a large variety of regular weak reflections with a transonic shock. To answer these questions, techniques from nonlinear elliptic theory are crucial.

(Sept. 18) C. Xie Title: Steady Subsonic Euler Flows through Two Dimensional Nozzles ---
In this talk, I will discuss existence, uniqueness, fine properties of two dimensional steady Euler flows through infinitely long nozzles. The one of the key points is to introduce the stream function, which can reduce Euler system into a single equations. When studying the equation for stream function, various elliptic estimates and blow up argument are involved.

(Sept. 25) J. Smoller Title: The Penrose Process: Extraction of Energy From a Rotating Black Hole ---
In 1969 Roger Penrose proposed a "gedanken" experiment in which energy can be extracted from a rotating black hole. People then showed numerically, or via asymptotic expansions that this is possible. We give a more convincing argument, at the level of the Cauchy problem, proving that this can be made rigorous. The method is to send (finely tuned, of course) "wave packet" initial data towards the black hole and study the evolution of this data. The main tool is to use our previously derived integral representation for the solution, together with a new result which implies that the outgoing energy is finite.

(Oct. 2) L. Bieri Title: An Extension of the Stability Theorem of the Minkowski Space in General Relativity ---
The talk addresses the global, nonlinear stability of solutions of the Einstein equations in General Relativity. In particular, it deals with the initial value problem for the Einstein vacuum equations, generalizing the results of D. Christodoulou and S. Klainerman in 'The global nonlinear stability of the Minkowski space'. Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is not bounded in $L^{\infty}$ any more. The main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime.

(Oct. 9) P. Kessenich Title: Global Existence for a 3D Incompressible Isotropic Viscoelastic Material ---
Incompressible viscoelastic materials can be studied using the Oldroyd-B model which views the material as an elastic polymer immersed in a Newtonian fluid. Previous global existence results for this model have used parabolic methods in which the size of the initial disturbance is inevitably dependent on the Newtonian viscosity in the Oldroyd-B equations. This talk will focus on the use of hyperbolic methods to prove global existence for these equations so that the smallness of the initial data is not restricted by the viscosity parameter.

(Oct. 16) M. Visan Title: Nonlinear Schrodinger equations at critical regularity ---
We introduce the nonlinear Schrodinger equation (NLS) and define criticality. We then survey the history of the two most studied cases of critical NLS, namely, the mass-critical NLS and the energy-critical NLS. This includes recent joint work with Rowan Killip, Terry Tao and Xiaoyi Zhang.

(Oct. 30) N. Jiang Title: Some recent progress on the hydrodynamic limits from Boltzmann equation ---
In this talk we address two topics: 1, In the framework of DiPerna-Lions renormalized solutions to the Boltzmann equation, we justify the incompressible Navier-Stokes limit in a bounded domain. Using the dissipative effects of the boundary layers (Knudsen layer and viscous fluid layer), the convergence is shown to be strong. 2, In the framework of classical solutions, We revisit the classical work of Caflisch for compressible Euler limit of the Boltzmann equation. By using a new L^{2}\mbox{-}L^{\infty} method, we prove the validity of the Hilbert expansion before shock formations in the Euler system with moderate temperature variation.

(Nov. 6) N. Masmoudi Title: Boundary layers and Homogenization ---
The goal of the talk is to discuss two related problems. The first one is the homogenization in polygonal domains and the second is about the derivation of the Navier boundary condition using the theory of homogenization.
Getting higher order corrections in the theory of (periodic) homogenization requires a very good understanding about what is happening at the boundary. This leads to boundary layer problems that can be solved using a quasi-periodic setting.
For the second problem, we consider the Navier-Stokes equation in a domain with rough boundaries. The small irregularity is modeled by a small amplitude and small wavelength boundary with typical lengthscale $\eps << 1$. Using boundary layer theory, we prove that the high order approximation leads to a Navier boundary condition. This is a joint work with David Gerard-Varet and preprints are availabe at ftp://ftp.dma.ens.fr/pub/reports/dma-08-14.pdf ftp://ftp.dma.ens.fr/pub/reports/dma-08-13.pdf

(Nov. 13) J. Rauch Title: Dispersive Stabilization ---
Quasilinear first order systems of partial differential operators defining ill posed initial value problems may be stabilized, that is turned into well posed initial value problems, by adding {\bf nonscalar} dispersive terms of Schr\"odinger type. In nonlinear optics such equations describe common instabilities. The stabilization is related to Turing's instability where stable equilibria can be destabilized by the addition of a nonscalar diffusive term. Research with Guy METIVIER.

(Nov. 20) I. Nenciu Title: On confining potentials and essential self-adjointness for Schroedinger operators on bounded domains ---
Let $\Om$ be a bounded domain in $\IR^n$ with $C^2$-smooth boundary, $\partial\Om$, of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on $\Om$ with potential $V \in L^{\infty}_{loc}(\Om )$. We seek the weakest conditions we can find on the rate of growth of the potential $V$ close to the boundary $\partial\Om$ which guarantee essential self-adjointness of $H$ on $C_0^\infty(\Om)$ . As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$ where $d(x)=\text{dist}(x,\partial\Om)$. More precisely, we show that if, as $x$ approaches $\partial\Om$, $$ V(x)\geq \frac{1}{d(x)^2}\biggl(\frac34-\frac{1}{\ln(d(x)^{-1})}- \frac{1}{\ln(d(x)^{-1})\cdot\ln\ln(d(x)^{-1})}-\cdots\biggr) $$ where the brackets contain an arbitrary finite number of logarithmic terms, then $H$ is essentially self-adjoint on $C_0^\infty(\Om)$. The constant 1 in front of each logarithmic term is optimal. The proof is based on a refined exponential Agmon estimate combined with a well known multidimensional Hardy inequality.

(Dec. 4) C. Bardos Title: Multiconfiguration Time Dependent Hartree Fock Method ---
This talk is devoted to the so called MCTDHF method for the approximation of solutions of $N$ particles Schr\"odinger equation with binary iteraction. With the Pauli exclusion principle a natural Galerkin basis for approximation would be a time dependent linear fintie sum of Slatter determinants constructed from a one particles hilbertian basis.
In the MCTDHF one considers also that such basis is time dependent. Therefore the solution is represented as living on the basis of a fiber bundle. Hence several representation for solutions among them one based on a variationnal principle and an other one called working equation more adapted to theoretical and numerical analysis.
The introduction of the variationnal equation leads to the conservation of energy and to some a posteriori estimates.
Using both representation one obtain local and global existence and stability results for the solution.
Eventually a the invertibility of a global ``density" matrix plays a crucial role in the analysis and in the computation. Regularising this matrix destroy the conservation of energy and therefore requires for global stability an analysis in less regular spaces. Such analysis is provided by standard use of Strichartz estimates.