Applications of optical mass transport theory to partial differential equations and to geometric inequalities

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My research focuses on applications of Optimal transportation theory to partial differential equations arising in kinetic theory, and to geometric inequalities, and it consists of the following three parts: 1. Spatially inhomogeneous kinetic equations. These equations describe the motion of particles undergoing diffusion, transport and collision in a surrounding bath. They are used as models for the time evolution of granular media, of electrons in a laser light, or ions in a superionic conductor. The mathematical equations are the  kinetic Fokker-Planck equations (e.g. Vlasov-Poisson-Fokker-Planck system, Boltzmann equation), and the granular media equations. Techniques from Optimal transportion theory are expected to lead to a better understanding of these equations. The purpose of this research is to use tools from Optimal transportation theory to study the global existence, uniqueness and large time behavior of solutions of these equations, and in particular, derive the -sharp- rates at which these solutions converge to an equilibrium, in case there is one. 2. Parabolic diffusion equations. Typically, these are the spatially homogeneous equations associated to the ones mentioned above (e.g. Fokker-Planck type equations, p-Laplacian equations, doubly degenerate equations). We plan to investigate whether the p-Laplacian and the doubly degenerate equations are gradient flows with respect to an optimal transportation differential structure (as known already for the spatially homogeneous Fokker-Planck type equations), and to find the -sharp- rates of convergence to equilibria of the solutions of these equations. 3. Geometric inequalities. Best constants and extremals in geometric inequalities (e.g. Logarithmic-Sobolev inequality) can lead to sharp rates of decay to equilibria for evolution equations (e.g. linear Fokker-Planck equation). We intend to use Optimal transportion theory to find the best constants and extremals of all the Gagliardo-Nirenberg inequalities, and apply them to study large time asymptotics of evolution equations.