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

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 ${\mathbb{R}}^3$. 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 $O(t^{-3/2+\frac{3}{2r}})$ in $L_{\xi }^2(L_x^r)$-norm for any $2\le r\le \infty$ 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.

42 pages, updated the file according to the many helpful and valuable comments of the referee

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