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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.

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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