本文考虑 Vlasov-Poisson-Fokker-Planck (VPFP) 方程和 Vlasov-Poisson-Landau (VPL) 方程的初值问题, 给出了在平衡态附近的线性化的 VPFP 方程和 VPL 方程的谱分析和半群估计, 并且给出了非线性问题解的最佳衰减速度. 本文证明当初值是整体 Maxwell 的小扰动时, VPFP 方程的解以指数衰减速度收敛到平衡态, 而 VPL 方程的解以代数速度 (1 + t)^−1/4 收敛到平衡态.

In this paper, we consider the initial value problems for Vlasov-Poisson-Fokker-Planck (VPFP) equations and Vlasov-Poisson-Landau (VPL) equations. We give the spectrum analysis and semigroup estimates on the linearized VPFP equations and VPL equations around their equilibrium states and show the optimal convergence rates of global solution to nonlinear problems. We show that the solution to VPFP equations tends to the equilibrium state at the exponential convergence rate, the solution to VPL equations tends to the equilibrium state at the algebraic convergence rate (1+t)^−1/4 when initial values are small perturbations of global Maxwellian.