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On the Vlasov-Maxwell System : régularity and non-relativistic limit

Abstract : In this dissertation, we study the Vlasov-Maxwell system of partial differential equations, describing the evolution of the distribution function of charged particles in a plasma. More precisely, we study the regularity of solutions to this system, and the question of the non-relativstic limit.In the first part, we study a Toy-model, combining the Vlasov equation with a system of transport equations. We use the methods developed to obtain and imrpove the Glassey-Strauss criterion, which gives a sufficient condition under which strong solutions do not develop singularities. The loss of regularity occures when the speed of the particles is close to the characteristic speed of the joined hyperbolic system. The same phenomenon occures for the solutions of the Toy-model, but its structure is easier to handle.In the second part, we focus on the question of the non-relativistic limit. After a rescaling of the equations, the speed of light can be considered as a big parameter. When it tends to infinity, it is called the non-relativistic limit. At first order, the non-relativistic limit of the Vlasov-Maxwell system is the Vlasov-Poisson system. First, an iterative method giving arbitrary high non-relativistic approximations is established. These systems combine the Vlasov-equation with elliptic systems of equations, and are well-posed in some weigthed Sobolev spaces. We also prove a result on the non-relativistic limit to the Vlasov-Poisson system under the weaker assumption of boundedness of the macroscopic density. We study a functional quantifying the Wasserstein distance between weak solutions of both systems.
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Nicolas Brigouleix. On the Vlasov-Maxwell System : régularity and non-relativistic limit. Analysis of PDEs [math.AP]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX098⟩. ⟨tel-03166920⟩

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