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Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. (English) Zbl 0619.35087
We propose a deterministic proof of the existence of global in time smooth solutions for the Vlasov-Fokker-Planck equations. The method relies on direct estimates of the decay of the solution when the velocity goes to infinity. It also yields a proof of the convergence of the solutions, towards those of the Vlasov-Poisson equation, when the diffusion coefficient goes to zero.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
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