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

152 p.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

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

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.MTM2014-52347-C2-1-R of DGE

Faster-than-c signals, special relativity, and causality

Motivated by the recent attention on superluminal phenomena, we investigate the compatibility between faster-than-c propagation and the fundamental principles of relativity and causality. We first argue that special relativity can easily accommodate -- indeed, does not exclude -- faster-than-c signalling at the kinematical level. As far as causality is concerned, it is impossible to make statements of general validity, without specifying at least some features of the tachyonic propagation. We thus focus on the Scharnhorst effect (faster-than-c photon propagation in the Casimir vacuum), which is perhaps the most plausible candidate for a physically sound realization of these phenomena. We demonstrate that in this case the faster-than-c aspects are ``benign'' and constrained in such a manner as to not automatically lead to causality violations.Comment: Plain LaTeX2E; 25 pages; 4 embedded figures (LaTeX pictures). V2: Some discussion clarified, minor rearrangements, references updated, no change in physics conclusions. To appear in Annals of Physic