Measure-valued weak solutions to some kinetic equations with singular kernels for quantum particles
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In this thesis, we present a mathematical study of three problems arising in the kinetic theory of quantum gases. In the first part, we consider a Boltzmann equation that is used to describe the time evolution of the particle density of a homogeneous and isotropic photon gas, that interacts through Compton scattering with a low-density electron gas at non-relativistic equilibrium. The kernel in the kinetic equation is highly singular, and we introduce a truncation motivated by the very-peaked shape of the kernel along the diagonal. With this modified kernel, the global existence of measure-valued weak solutions is established for a large set of initial data. We also study a simplified version of this equation, that appears at very low temperatures of the electron gas, where only the quadratic terms are kept. The global existence of measure-valued weak solutions is proved for a large set of initial data, as well as the global existence of $L^1$ solutions for initial data that satisfy a strong integrability condition near the origin. The long time asymptotic behavior of weak solutions for this simplified equation is also described. In the second part of the thesis, we consider a system of two coupled kinetic equations related to a simplified model for the time evolution of the particle density of the normal and superfluid components in a homogeneous and isotropic weakly interacting dilute Bose gas. We establish the global existence of measure-valued weak solutions for a large class of initial data. The conservation of mass and energy and the production of moments of all positive order is also proved. Finally, we study some of the properties of the condensate density and establish an integral equation that describes its time evolution.