Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

✍ Scribed by B. Perthame

• Publisher: John Wiley and Sons
• Year: 1990
• Tongue: English
• Weight: 459 KB
• Volume: 13
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670130508

✦ Synopsis

Abstract

Let us consider a solution f(x,v,t)ϵ__L__^1^(ℝ^2N^ × [0,T]) of the kinetic equation
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where |v|^α+1^ f~o~,|v|^α^ ϵL^1^ (ℝ__2N__ × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K~x~, we have
equation image
We use this result to improve the regularity of the local density ρ(x,t) = ∫ƒdν for the Vlasov–Poisson equation, which corresponds to g = __E__ƒ, where E is the force field created by the repartition ƒ itself. We also apply this to the Bhatnagar‐Gross‐;Krook model with an external force, and we prove that the solution of the Fokker‐Pianck equation with a source term in L^2^ belongs to L^2^([0, T]; H^1/2^(ℝ)).