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Optimal time decay of the Vlasov-Poisson-Boltzmann system in $${\mathbb R^3}$$. (English) Zbl 1232.35169
This article investigates the time convergence rate of solutions of the Vlasov-Poisson-Boltzmann system towards the Maxwellian equilibrium. The authors show an optimal $$\langle t\rangle^{-1/4}$$ convergence rate which is slower than the $$\langle t\rangle^{-3/4}$$ one for the Boltzmann equation because of driven forces. However, this rate is improved to $$\langle t\rangle^{-3/4}$$ for some well-prepared initial data.
The proof of this result is done in three steps: first, the authors perform a decomposition of the solution involving a few moments of the Maxwellian (macroscopic part) and a remainder (microscopic part). Next, they get a crucial $$L^2(\mathbb{R}_{\xi}^n)$$ time dependent estimate for the linearized system which leads to the $$\langle t\rangle^{-1/4}$$ convergence rate for the macroscopic part and the $$\langle t\rangle^{-3/4}$$ convergence rate for the microscopic part. Finally, using an energy estimate for the whole system, these estimates are shown to still hold for the nonlinear system.

##### MSC:
 35Q83 Vlasov equations 35Q20 Boltzmann equations 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs
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