  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.
