On Lorentz Invariance, Regularity and Global Existence for Kinetic Equations

Doctoral thesis, 1997

[A1] Regularity of the gain term and strong L1 convergence to equilibrium for the relativistic Boltzmann equation. The main purpose of the paper is to show that the gain term of the relativistic collision operator is regularizing. This is a generalization of P.L. Lions' analogous result in the nonrelativistic situation. The Lorentz invariance of relativistic particle dynamics plays an important role in the proof. The regularizing theorem has many applications in kinetic theory, some of which are discussed in the paper. In particular the asymptotic behaviour of periodic solutions to the relativistic Boltzmann equation is studied. It is shown that such solutions converge strongly in L^1 to a global Juttner equilibrium solution for arbitrary initial data. This extends earlier results for small data. [A2] Global existence of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzman equation. This paper shows global existence and uniqueness of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzmann equation. This extends an earlier result from the two-dimensional situation. Moreover, assumptions on the scattering kernel which limit the decay of the collision frequency are introduced, extending in part the assumptions used in the two-dimensional case. A slow decay of the collision frequency is important for studying fluid approximations under high electric fields of the Vlasov-Poisson-Boltzmann equation. [A3] Controlling the propagation of the support for the relativistic Vlasov equation with a selfconsistent Lorentz invariant field. The motivation for this paper comes from the fact that the successful techniques developed for controlling the propagation of the support for the classical Vlasov-Poisson equation (leading to global existence of smooth solutions) all fail for the relativistic Vlasov-Poisson equation. This equation lacks the Lorentz invariance. It has nevertheless been suggested that an understanding of this equation may be necessary for understanding the fundamental relativistic Vlasov-Maxwell equation. In the paper a new equation for the field is introduced which is Lorentz invariant. It is shown that the propagation of the support, for solutions satisfying this equation and the relativistic Vlasov equation, may be controlled. This is a strong indication that the transformation properties of the equations are important in studying existence of smooth solutions.