35F20 Nonlinear firstorder equations
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Existence Results for Plasma Physics Models Containing a Fully Coupled Magnetic Field
(2009)

Martin Seehafer
 The present thesis concern is the initial value problem for three nonlinear systems of partial differential equations: the VlasovDarwin system, the VlasovPoisswell system and a version of the latter which is called the modified VlasovPoisswell system. These equations belong to kinetic theory, which has proved useful when describing large particle systems in different areas of physics such as kinetic theory of gases, the formation of stellar structures or plasma physics. In the present thesis equations originating in plasma physics are considered which describe the evolution of the time dependent density function f(t,x,v) (t  time, x – position, v  particle velocity) of a large ensemble of charged particles in the (x,v)phase space influenced by the electromagnetic field created by the particles and when neglecting collisions. The focus of the investigation is on existence and uniqueness questions for solutions of the initial value problem, i.e., it is asked whether there exists a solution f of the system under consideration such that f(t=0)=f0 where f0 is a prescribed initial datum. In order to answer this question further properties of solutions such as energy and charge conservation or decay rates must be taken into account. An important issue is, whether  if necessary under additional hypotheses or by weakening the concept of solution  global solutions, i.e., solutions existing for all t>=0, may be obtained. The most important results are a theorem about local existence and uniqueness of classical solutions of the VlasovPoisswell system, a global existence result for weak solutions of the modified VlasovPoisswell system, and a global existence theorem for classical solutions of the VlasovDarwin system under the assumption of smallness of the initial.