Measure-valued weak solutions for some kinetic equations with singular kernels for quantum particles.

Abstract

In this thesis, we present a mathematical study of three problems arising in the kinetic theory of quantumgases.In the first part, we consider a Boltzmann type equation that is used to describe the evolution of theparticle density of a homogeneous and isotropic photon gas, that interacts through Compton scatteringwith a low-density electron gas at non-relativistic equilibrium.Due to the highly singular redistribution function, we consider an approximation that is, nevertheless, stillsingular at the origin. The global existence of measure-valued weak solutions for a large set of initial datais established.We also study a simplified version of this equation, that appears at very low temperatures of the electrongas, where only the quadratic terms are kept. The global existence of measure-valued weak solutions isproved for a large class of initial data, as well as the global existence of $L^1$ solutions for initial datathat satisfy a strong integrability condition. The long time asymptotic behavior of weak solutions for thissimplified equation is also described.In the second part of the thesis, we consider a system of two coupled kinetic equations related to ansimplified model for the evolution of the particle density of the normal and superfluid components in ahomogeneous and isotropic weakly interacting dilute Bose gas.We prove the global existence of measure-valued weak solutions for a large class of initial data. Theconservation of mass and energy and the production of moments of all positive order is also established.Finally, we study some of the properties of the condensate density and we establish an integral equationthat describes its time evolution.