The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials

TL;DR

This paper proves the global existence of classical solutions near Maxwellians for the Vlasov-Maxwell-Boltzmann system with very soft potentials in the whole space, overcoming previous challenges related to degenerate dissipation and regularity loss.

Contribution

It establishes the first comprehensive global existence result for the system with very soft potentials, addressing complex nonlinear and regularity issues.

Findings

Proves global existence of solutions near Maxwellians.

Handles the degenerate dissipation at large velocities.

Overcomes regularity-loss challenges in electromagnetic fields.

Abstract

Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft potentials has been an open problem. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper aims to resolve this problem in the whole space provided that initial perturbation has sufficient regularity and velocity-integrability.