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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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