The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
In this talk, I will discuss some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings, with Dirichlet-type boundary conditions. A solution is called quasiconcave if its superlevel sets are convex. I will present two situations for which the solutions are not quasiconcave. This talk is based on a joint work with N. Nadirashvili and Y. Sire.
The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we show how the recently developed theory of hypocoercivity can be used to prove exponentially fast convergence to equilibrium for such diffusion processes. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.
If the pieces of differential graded algebra hold together as they should, one obtains a tower of finite dimensional ODE's related by semiconjugacies [i.e., maps that commute with time evolution] that are derived from the Navier Stokes PDE in 3D -- which is itself used to model incompressible Newtonian fluid motion. Using these ODEs one can show the existence of measures at the top of the tower which project to consistent stationary measures for the ODE of each model in the tower. One can calculate algebraically the ODEs by algorithms that depend only on the combinatorics of cellular decompositions. These ODEs can be treated numerically to compute stationary measures.
I'll describe the asymptotic behavior of solutions to a reaction-diffusion equation with KPP-type nonlinearity. One can interpret the solution in terms of a branching Brownian motion. It is well-known that solutions to the Cauchy problem may behave asymptotically like a traveling wave moving with constant speed. On the other hand, for certain initial data, M. Bramson proved that the solution to the Cauchy problem may lag behind the traveling wave by an amount that grows logarithmically in time. Using PDE arguments, we have extended this statement about the logarithmic delay to the case of periodically varying reaction rate. We also consider situations where the delay is larger than logarithmic. This new PDE approach involves the study of the linearized equation with Dirichlet condition on a moving boundary. This is joint work with Francois Hamel, Jean-Michel Roquejoffre, and Lenya Ryzhik.
We will discuss fundamental analytical tools for the classical non-linear Boltzmann equation. The main focus will lay in the the interplay of properties of the collision kernels associated to the collision operator and the generation and propagation of summability of moments of the solution for the homogeneous initial value problem. Such summability yields global bounds for the solution of the Boltzmann equation by exponentially weighted norms in L^1 and pointwise, where the exponent depend on the initial state norms, the rate of the intra-molecular potentials as well as the integrability properties on the sphere (angular averaging) for the scattering angle cross-section. The study these angular averaging properties play a fundamental role in the construction of exponentially weighted bounds.