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Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. (English) Zbl 1120.35016
The author studies the linear Fokker-Planck equation in \(\mathbb{R}^{2d}_{x,v}\), \(d\geq 3\), \[ \partial_t f + v\cdot \partial_x f- \partial_x V\cdot \partial_v f - \gamma \partial_v\cdot(\partial_v +v)f =0, \] where \(f\) is a given external confining potential such that \(V\geq 0\), \(V'' \in W^{\infty,\infty}\), and \(e^{-V}\) is a fast decaying function; moreover \(\gamma\) is a positive constant and \(f\) is a distribution function of particles. The equation admits a unique normalized Maxwellian \(\mathcal{M}\), and the analysis is carried out in the space \(B^2 = \mathcal{M}^{1/2}L^2\). The author proves estimates for the semigroup generated by \(-K = -v\cdot \partial_x f+ \partial_x V\cdot \partial_v f + \gamma \partial_v\cdot(\partial_v +v)f\) and its derivatives as well as estimates determining the rate of convergence of this semigroup to \(\mathcal{M}\) as \(t\to \infty\) (under additional assumption on the spectral gap for the Witten Laplacian associated with \(K\)). These results are used to prove global solvability of the nonlinear mollified Vlasov-Poisson-Fokker-Plank equation and to provide estimates of the rate of decay of the relative entropy associated with the latter equation. Mollification here refers to the fact that the potential is given by a ‘smeared’ solution of the self-consistent Poisson equation.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
47D06 One-parameter semigroups and linear evolution equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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