Global smooth dynamics of a fully ionized plasma with long-range collisions

Abstract

The motion of a fully ionized plasma of electrons and ions is generally governed by the Vlasov–Maxwell–Landau system. We prove the global existence of solutions near Maxwellians to the Cauchy problem of the system for the long-range collision kernel of soft potentials, particularly including the classical Coulomb collision, provided that both the Sobolev norm and $L_{\xi }^2(L_x^1)$-norm of initial perturbation with enough smoothness and enough velocity weight is sufficiently small. As a byproduct, the convergence rates of solutions are also obtained. The proof is based on the energy method through designing a new temporal energy norm to capture different features of this complex system such as dispersion of the macro component in ${\mathbb{R}}^3$, singularity of the long-range collisions and regularity-loss of the electromagnetic field.