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The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials. (English) Zbl 1414.82027
This paper solves the problem of the global existence of perturbative classical solutions around a global Maxwellian to Vlasov-Maxwell-Boltzmann (VMB) systems in the whole space, provided that the initial perturbation has sufficient regularity and velocity-integrability. This is done thanks to the proposed approach where the global classical solutions to the VBM system for the whole range of soft potentials are given. The obtained result shows that, as long as the initial data is small with enough regularity, one can establish the global existence of small amplitude classical solutions for the full range of cutoff intermolecular interactions with $$-3 < \gamma\le 1$$.
The paper is divided into four main sections. In the introduction, we have the scope with the information about VMB systems like dilute ionized plasmas consisting of two-species particles. Section 2 gives necessary theorems and explanations about difficulties in the case $$-3 < \gamma\le 1$$. It shows the main results; they are given in the following steps:
explanations about existing approaches with the list of main difficulties: (i) how to control the possible velocity-growth, (ii) how to control the convection term $$\nu \cdot \nabla_x f$$ in the weighted energy estimates.
the case of soft potentials, $$-1 \le\gamma \le 1$$.
presentation of the authors’ approach.
Section 3 shows all necessary proofs for the main results. Section 4 gives proofs of lemmas and estimates used in Section 3.

##### MSC:
 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35Q83 Vlasov equations 35Q61 Maxwell equations 35Q20 Boltzmann equations 35L60 First-order nonlinear hyperbolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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