Existence Theory for the Landau System from Plasma Physics.
The Landau System is a classical model in the Kinetic theory of plasma. This system was derived by L. D. Landau in 1963 from the Boltzmann system by taking advantage of the long range of the Coulomb force. It is generally believed that the Landau system can be used to describe a plasma statistically when the collisions between the moving particles(soft collisions) are long range. After Landau derived the system, many efforts have been devoted to study the system. The first natural question one would ask about the system is the following initial value problem: Can this system admit a solution under any given initial distribution of moving particles? Due to the intrinsic difficulty associated with the system, few results have been obtained on this topic. In my thesis, I intend to establish a systematic study on the existence problem, especially the initial value problem aforementioned. I have obtained existence of local (in time) solutions for both Landau-Maxwell and Landau-Poisson systems. Furthermore, some result about the regularity of global solutions has been obtained and the relation between the Landau system and Vlasov system has been investigated. We proved that under certain condition, the solutions of the Landau system will converge to the solution of the Vlasov system as the collision frequency approaches zero. Various methods and techniques have been employed which including, the iteration method and the Bernstein Method. The most recent results on the smoothing effect of the velocity averages played an important role in obtaining the results in the thesis. A result concerning the Landau damping of the linearized relativistic Vlasov Equations is also presented in the thesis. We proved that for a set of initial data and governing functions, the Landau damping phenomena will not occur in the linearized relativistic Vlasov Equations.
- Pub Date:
- January 1995
- Mathematics; Physics: Elementary Particles and High Energy; Physics: Fluid and Plasma