The Cauchy Problem in the Kinetic Theory of Hot Dilute Plasmas

The relativistic Vlasov-Maxwell (RVM) system is a kinetic model that
describes the evolution of a collisionless plasma whose particles interact
through the self-induced electromagnetic field. This model is essential in
the study of hot dilute plasmas, where collisions can be neglected and the
particles can move at speeds comparable to the speed of light. In the full
three dimensional space, 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. The hyperbolic nature of the
Maxwell equations and the nonlinear coupling with the transport Vlasov
equation amount to the challenges imposed by this system. If the particle
interaction is assumed to be a second-order relativistic correction to the
full Maxwell case, then the RVM reduces to the Vlasov-Darwin system. A
consequence of this assumption is that instead of the less tractable
hyperbolic Maxwell equations, the resulting system has elliptic features
even though there is a fully coupled magnetic field. As for the RVM,
global in time existence of smooth solutions without restriction in size
on the Cauchy data remains an open problem for the Vlasov-Darwin system as
well.

In this talk I will discuss some of the key results toward the solution of
the Cauchy problem for these two systems. I will start with earlier work
leading up to recent contributions I have made in collaboration with R.
Illner and M. Agueh.