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

In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space ${\mathbb{R}}^3$. 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 $O(t^{-3\mathrm{/}2+\frac{3}{2r}})$ in $L_{\xi }^2(L_x^r)$-norm for any $2\le r\le \mathrm{\infty }$ 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.