**The seminar meets regularly on Wednesdays** at **4pm** in **Eckhart 202**. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Will Feldman.

Gradient flow from particle models

This talk explains how a large class of nonlinear diffusion equations can be expressed as gradient flows with respect to a special Lyapunov functional, the thermodynamic entropy, and a special metric, the thermodynamic metric. These quantities give information beyond the limit PDE, they capture the large deviations and fluctuations as well. We will also describe methods to extract the metric (and entropy differences) from numerical simulations of an interacting particle system.

This is joint work with Peter Embacher (Cardiff), Celia Reina (UPenn), Marios Stamatakis (Bath) and Johannes Zimmer (Bath).

Stochastic homogenisation for degenerate Hamilton-Jacobi equations

In the talk I Investigate the limit behaviour for a family of Cauchy problems for Hamilton-Jacobi equations describing a stochastic microscopic model. The Hamiltonian considered is not coercive in the total gradient. The Hamiltonian depends on a lower dimensional gradient variable which is associated to a Carnot group structure. The rescaling is adapted to the Carnot group structure, therefore it is anisotropic. Under suitable stationary-ergodic assumptions on the Hamiltonian, the solutions of the stochastic microscopic models will converge to a function independent of the random variable: the limit function can be characterised as the unique viscosity solution of a deter- ministic PDE. The key step will be to introduce suitable lower-dimensional constrained variational problems.

Global well-posedness for the 2D Muskat problem with slope less than 1

The Muskat problem was originally introduced by Muskat in order to model the interface between water and oil in tar sands. In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media. In this talk I will prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Naverov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.

Special Quasirandom Structures: a selection approach for stochastic homogenization

We consider the homogenization of linear elliptic PDEs with random stationary coefficients. As is well-known, the deterministic homogenized coefficients are obtained through a corrector problem that is set on the entire space. This problem is thus intractable from the practical viewpoint. A standard approximation consists in considering large but bounded domains, and solve the corrector problem (complemented with appropriate boundary conditions) on these domains. The obtained apparent effective coefficients are random. It is thus natural to consider several realizations of the microstructure, in a Monte Carlo fashion.

In this talk, we describe a selection method, where we a priori select the microstructures for which we solve the corrector problem. This selection is performed on the basis of some appropriate criteria. In that spirit, the expensive corrector problem is solved only for microstructures that are ``more representative'' than generic microstructures of the materials on the entire space. We will discuss this approach, both from a theoretical and numerical standpoints.

Joint work with C. Le Bris and W. Minvielle.

From the Newton equation to the wave equation in case of shocks

It has been recently proved by X. Blanc, C. Le Bris and P.-L. Lions that one can derive the wave equation from a chain of numerous particles submitted to the Newton equations (each particle interacting through a potential with its nearest neighbours). However, the assumptions necessary for this derivation forbid any shock wave, whereas nonlinear wave equations tend to create shocks for almost every smooth initial data. Therefore, with X. Blanc, we tried to understand the specific phenomena involved in shock, for linear and non-linear potentials. Surprisingly enough, we discovered that in the linear case, the Newton equations still tend to the wave equation whether there are shocks or not, but that this is false for a large class of non-linear convex potentials as soon as there is a shock: in this case, one can observe dispersive shock waves (which is well-known in the case of Toda lattice). Thus, in that non-linear convex cases, an atomic chain submitted to the Newton equations can be well described by the wave equation until a shock, and then this approximation becomes false. In this talk, I give a few ingredients of the proof, which requires anyway a conjecture unproved until now.

Approximation of fractional diffusion by transport equations: Recent developments

I will review some classical results and discuss recent developments in the derivation of fractional diffusion equations from kinetic models. In particular, I will present some new results concerning the role of external acceleration fields and the derivation of Neumann type boundary conditions from microscopic boundary reflection laws.

For questions, contact Will Feldman at: feldman