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Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space. (English) Zbl 1244.35010
Summary: 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 [R. M. Strain and Y. Guo, Commun. Partial Differ. Equations 31, No. 1–3, 417–429 (2006; Zbl 1096.82010)]. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work 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 $$O(t^{-3/2 + 3/(2r)})$$ in the $$L^2_\xi (L^r_x)$$-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.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35Q20 Boltzmann equations 35Q83 Vlasov equations 35Q61 Maxwell equations
##### Keywords:
optimal decay rate
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