Autumn 2013

Date Speaker Title/Abstract
10 Oct Mahir Hadzic
King's College London
Stability and decay in the classical Stefan problem
The Stefan problem is a well-known free boundary problem modeling phase transitions. I will survey some recent results on the well-posedness and stability theory in presence and absence of surface tension. I will then show a global stability result in absence of surface tension, thereby explaining a hybrid methodology combining high-order energy methods and quantitative Hopf-type lemmas.
24 Oct Daniel Marahrens
MPI Leipzig
Quantitative results in stochastic homogenization
Consider a discrete elliptic equation on the integer lattice with random coefficients, arising for example as the steady state of a diffusion through the lattice with random diffusivities. Classical homogenization results show that with ergodic coefficients and on large scales, the solutions behave as the solutions to a diffusion equation with constant homogenized coefficients. The homogenized coefficients can be characterised through the solution to a so-called 'corrector problem'. Recently Gloria, Otto and Neukamm have developed tools to obtain optimal estimates for the corrector problem via a spectral gap inequality. In this talk, I will present how to obtain optimal estimates on the discrete Green's function under the assumption of a logarithmic Sobolev inequality. I will indicate how to obtain optimal estimates on the homogenization error from these Green function estimates. This is joint work with Felix Otto.
4 Nov Mythily Ramaswamy
TIFR Centre for Applicable Mathematics, Bangalore
Control of Compressible Navier-Stokes System
After discussing the stabilizability of finite dimensional systems via feedback control, similar results in the PDE context, for Compressible Navier-Stokes system will be outlined.
7 Nov François Bolley
Paris Dauphine University
Long time behaviour for the granular media equation
The granular media equation is a non linear Fokker-Planck type equation, derived from the Boltzmann equation, which models the evolution of a set of particles subject to almost elastic collisions. Studying the long time behaviour of its solutions has recently raised much attention, by entropy dissipation techniques, based on its interpretation as a gradient flow in the space of probability measures, or by contraction properties in Wasserstein distance, also linked with the optimal transport problem. We will review these results, and present a new method, based on estimating the dissipation of this distance along the flow: this method enables to cover the physically relevant potentials and can hopefully be useful for other models. This is joint work with J. A. Carrillo (London) I. Gentil (Lyon) and A. Guillin (Clermont-Ferrand).
15 Nov Milton Lopes Jr
Federal University of Rio de Janeiro
Weak and wild solutions of the incompressible Euler equations
The theory of weak solutions of the incompressible Euler equations is seeing intense current activity, due to the introduction of ideas from differential geometry by C. De Lellis and L. Székelyhidi. Much of this activity is concerned with the construction, and the basic properties of "wild solutions", built with convex integration techniques. The purpose of this talk is to give a broad overview of the field, highlighting the impact of the ideas surrounding wild solutions, looking also at other recent developments and directions for future research.
21 Nov Matthias Liero
WIAS Berlin
On gradient structures for drift-reaction-diffusion systems and Markov chains
In this talk, I will show that reaction-diffusion systems and Markov chains satisfying a detailed balance condition can be formulated in a natural way as a gradient flow for the relative entropy. The gradient structures consist of a driving functional, e.g. the relative entropy of the system, and a so-called Onsager operator, which is an extension of the Riemannian theory developed by Otto and his coworkers for Wasserstein gradient flows. The central point is that in the Onsager form we have an additive splitting of the Onsager operator, e.g. into a diffusive and a reaction part. In particular, I will highlight the benefit of gradient structures and the Onsager operator for the modelling of semiconductor devices such as solar cells. In the second part of the talk I present some results concerning the geodesic lambda-convexity of the gradient structures.
5 Dec Qian Wang
University of Oxford
A vector field approach for sharp local well-posedness of quasilinear wave equations
We give a proof of the sharp local well-posedness of general quasilinear wave equation by a vector field approach. The $H^{2+\epsilon}$ local well-posedness result was proved by Smith and Tataru by constructing parametrix using wave packet. Based on the commuting vector field approach, this type of results have been established by Klainerman and Rodnianski for $(3+1)$ Einstein vacuum equations, by taking advantage of $\mathbf{Ric}=0$. However, $\mathbf{Ric}$ of the metric of the general type of equations contains second order derivatives of the metric, which makes it very difficult to implement the vector field approach to derive the sharp result. To get around this difficulty, our idea is to employ a conformal method to modify the spacetime metric locally such that the second order term in $\mathbf{Ric}(L,L)$ vanishes, with $L$ the tangential vector field of the null pair $\{L, \underline{L}\}$. The null hypersurface of the modified metric is then smooth enough to provide a bounded Morawetz type energy. The other difficulty we encounter is that we only have very limited control on derivatives of the conformal factor. This leads to worse behaviour of the deformation tensor ${}^{(K)}\pi$ of Morawetz vector field $K$ under the conformal metric. The part of ${}^{(K)}\pi$ coming from $\underline{L}$ in the Morawetz vector field becomes worse while the part coming from $L$ behaves better via the conformal change of the metric. If using the standard approach to derive Morawetz energy, these two parts will inevitably appear simultaneously. We solve this issue by adapting the new physical approach by Dafermos and Rodnianski to derive Morawetz energy by using vector fields involving $L$ only.
(This abstract is also available in PDF format.)