The Cauchy problem for the 3D relativistic Vlasov-Maxwell system and its Darwin approximation

The relativistic Vlasov-Maxwell system (RVM for short) is a kinetic model that arises in plasma physics and describes the time evolution of an ensemble of charged particles that interact only through their self-induced electromagnetic field. Collisions among the particles are neglected and they are assumed to move at speeds comparable to the speed of light. If the particles are allowed to move in the three dimensional space, then the main open problem concerning this system is to prove (or disprove) that solutions with sufficiently smooth Cauchy data do not develop singularities in finite time. Since the RVM system is essential in the study of dilute hot plasmas, much effort has been directed to the solution of its Cauchy problem. The underlying hyperbolic nature of the Maxwell equations and their nonlinear coupling with the Vlasov equation amount for the challenges imposed by this system. In this thesis, we show that solutions of the RVM system with smooth, compactly supported Cauchy data develop singularities only if the charge density blows-up in finite time. In particular, solutions can not break-down due to shock formations, since in this case scenario the solution would remain bounded while its derivative blows-up. On the other hand, if the transversal component of the displacement current is neglected from the Maxwell equations, then the RVM system reduces to the socalled relativistic Vlasov-Darwin (RVD) system. The latter has useful applications in numeric simulations of collisionless plasma, since the hyperbolic RVM is now reduced to a more tractable elliptic system while preserving a fully coupled magnetic field. As for the RVM system, the main open problem for the RVD system is to prove whether classical solutions with unrestricted Cauchy data exist globally in time. In the second part of this thesis, we show that classical solutions of the RVD system exist provided the Cauchy datum satisfies some suitable smallness assumption. The proof presented here does not require estimates derived from the conservation of the total energy nor those on the transversal component of the electric field. These have been crucial in previous results concerning the RVD system. Instead, we exploit the potential formulation of the model equations. In particular, the Vlasov equation is rewritten in terms of the generalized variables and coupled with the equations satisfied by the scalar and vector Darwin potentials. This allows to use standard estimates for singular integrals and a recursive method to produce the existence of local in time classical solutions. Hence, by means of a bootstrap argument, we show that such solutions can be made global in time provided the Cauchy data is sufficiently small.

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