Optimal Large-Time Behavior of the Vlasov-Maxwell-Boltzmann System in the Whole Space

TL;DR

This paper establishes the optimal large-time decay rates of solutions to the Vlasov-Maxwell-Boltzmann system in the whole space, showing convergence to Maxwellian equilibrium with explicit decay rates for the electromagnetic field.

Contribution

It proves the asymptotic decay rates of solutions to the Vlasov-Maxwell-Boltzmann system, extending previous results by providing explicit optimal decay rates in various norms.

Findings

Solutions converge to Maxwellian at rate O(t^{-3/2+3/(2r)}) in L^2_xi(L^r_x) norm.

Electromagnetic field tends to zero with explicit decay rates.

Global solutions exist and decay optimally under smooth, fast-decaying initial conditions.

Abstract

In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work [10] on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-3/2+\frac{3}{2r}})$ in $L^2_\xi(L^r_x)$-norm for any $2\leq r\leq \infty$ if initial perturbation is smooth enough and decays in space-velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided.