Abstract: In this talk I will discuss some recent results related to the large time behavior of first order models with pairwise attraction and repulsion. The first part concerns the Newtonian repulsion with confinement (joint work with Eitan Tadmor). We prove the uniqueness of steady states for convex radial confining potentials, and prove an algebraic equilibration rate by constructing a Lyapunov-type functional. The uniqueness result is extended to the attraction-repulsion case with Newtonian repulsion and near-quadratic attraction. The second part concerns general attraction-repulsion (joint work with Jose Carrillo). For potentials satisfying the linear interpolation convexity (LIC), we prove the radial symmetry of Wasserstein-$\infty$ local minimizers. With some further assumptions, we prove the uniqueness of Wasserstein-$\infty$ local minimizers. When applying to power-law potentials, it implies that the steady state constructed in Carrillo-Huang 16' is indeed the unique local minimizer.
Abstract: In this talk I will talk about a framework for studying approximate controllability in infinite dimensional spaces. As an application, I will talk about how this framework can be used to deduce the topological irreducibility of SPDE, which is closely related to the unique ergodicity. This talk is based on the work by Glatt-Holtz, Herzog and Mattingly.