Uniform Lifetime for Classical Solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell System

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function $f(t,\cdot)$. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field $x \mapsto \epsilon^{-1} \mathbf{B}_e(x)$, where $\epsilon$ is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields $(E,B)$ are supposed to be small. In the non-magnetized setting, local $C^1$-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since $\epsilon^{-1}$ is large, standard results predict that the lifetime $T_\epsilon$ of solutions may shrink to zero when $\epsilon$ goes to $0$. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound ($0<T<T_\epsilon$) on the lifetime of solutions and uniform Sup-Norm estimates. A bootstrap argument allows us to show $f$ remains at a distance $\epsilon$ from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.