Negative Sobolev Spaces and the Two-species Vlasov-Maxwell-Landau System in the Whole Space

TL;DR

This paper proves the global existence of solutions for the two-species Vlasov-Maxwell-Landau system near a Maxwellian in the whole space, using weaker regularity and smallness assumptions than previous work, and extends the approach to related systems.

Contribution

It introduces a novel method that does not depend on linear decay or Duhamel's principle, enabling broader applicability to related kinetic systems.

Findings

Established global solvability near Maxwellian

Weaker regularity and smallness conditions than prior work

Applicable to one-species Vlasov-Maxwell-Landau and Vlasov-Maxwell-Boltzmann systems

Abstract

A global solvability result of the Cauchy problem of the two-species Vlasov-Maxwell-Landau system near a given global Maxwellian is established by employing an approach different than that of [5]. Compared with that of [5], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov-Maxwell-Landau system for the case of $\gamma>-3$ and the one-species Vlasov-Maxwell-Boltzmann system for the case of $-1<\gamma\leq 1$ to deduce the global existence results together with the corresponding temporal decay estimates.