We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space $H^1(\mu )$ with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

中文翻译：

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具有平均场相互作用的动力学Fokker-Planck方程

我们研究具有平均场相互作用的动力学Fokker-Planck方程的长时间行为，该方程的极限通常称为Vlasov-Fokker-Planck方程。我们证明了加权Sobolev空间中溶液的均匀（按粒子数量计）指数收敛到平衡$H^{\mathrm{1个}}（\mu ）$收敛速度可以明确计算，并且与粒子数无关。证明的独创性取决于Lyapunov类型条件下的功能不等式和矫顽力，通常不适合提供三维结果。