Solutions to the Fokker-Planck equation with time-dependent force and source in $\mathbb R^3$

Abstract

This paper concerns the global solutions of the kinetic Fokker-Planck (KFP) equation in $\mathbb R^3$ driven by a given external force $E=E(t,x)$ and an inhomogeneous source term $S_0=S_0(t,x,v)$. First, when $E$ and $S_0$ are sufficiently small in a suitable hybrid homogeneous Besov-Sobolev framework without any time-decay assumption, we establish the global well-posedness of the Cauchy problem in the hybrid homogeneous space $L_v^2(\dot B^{ {1}/{2}}_{2,\infty}(\mathbb R^3_x) \cap\dot H^N(\mathbb R^3_x))$ with $N\geq 3$. We further obtain the uniform stability of the Cauchy problem via semi-group theory and a refined energy method. Second, when both $E$ and $S_0$ are time-periodic of the period $T$, we show the existence of time-periodic solutions with the same period $T$ and their asymptotic stability as $t\to \infty$. Moreover, we derive the time-decay estimate of the perturbation around time-periodic solutions for initial data in $L^p$-norm with $ {3}/{2}< p \leq 2$. Finally, when both $E$ and $S_0$ are independent of $t$, we establish the existence of stationary solutions and their asymptotic stability. Our results provide a unified treatment of time-periodic dynamics and stationary states for the KFP equation with small force and source terms.