Weak Stability and Large Time Behavior for the Cauchy Problem of the Vlasov-Maxwell-Boltzmann Equations

TL;DR

This paper investigates the weak stability and long-term behavior of solutions to the Vlasov-Maxwell-Boltzmann equations, demonstrating convergence to Maxwellian distributions over time.

Contribution

It establishes the weak stability of renormalized solutions and analyzes their large time asymptotics, overcoming difficulties related to partial differentiability.

Findings

Weak stability of renormalized solutions is proven.

Solutions tend to a local Maxwellian as time approaches infinity.

The analysis uses compactness of velocity averages and renormalized formulations.

Abstract

The Cauchy problem for the Vlasov-Maxwell-Boltzmann equations (VMB) is considered. First the renormalized solution to the Vlasov equation with the Lorentz force is discussed and the difficulty on the partial differentiability of the coefficients is overcome. Then the weak stability of the renormalized solutions to the Cauchy problem of VMB is established using the compactness of velocity averages and a renormalized formulation. Furthermore, the large time behavior of the renormalized solutions to VMB is studied and it is proved that the density of particles tends to a local Maxwellian as the time goes to infinity.