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Global smooth dynamics of a fully ionized plasma with long-range collisions. (English) Zbl 1305.82057
The author describes the dynamics of a fully ionized, two-component plasma under the influence of the self-consistent Lorentz force and binary collisions (taken into account by the Landau collision operator) by a system of kinetic transport equations and the Maxwell equations, which he calls “Vlasov-Maxwell-Landau system”. The main aim of the paper is the prove the global existence of solutions of the Cauchy problem of the Vlasov-Maxwell-Landau system of equations in the form of a disturbed global Maxwellian velocity distribution under some conditions on the initial data. In doing so, the long-range collisional kernel of soft potentials, particularly including the classical Coulomb collisions, is taken into account – assuming that both the Sobolev norm and the $$L^2_\xi(L^1_x)$$-norm of the initial perturbation “with enough smoothness and enough velocity weight” are sufficiently small. As a byproduct, also the convergence rates of the solutions are obtained. The proof is based on the energy method through designing a new temporal energy norm to capture different features of the complex system such as dispersion of the macro component in the velocity space, singularity of the long-range collisions and regularity-loss of the electromagnetic field.

##### MSC:
 82D10 Statistical mechanics of plasmas 35Q84 Fokker-Planck equations 35Q83 Vlasov equations
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