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

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 $L_v^2({\dot{B}}_{2,\mathrm{\infty }}^{1\mathrm{/}2}({\mathbb{R}}_x^3)\cap {\dot{H}}^N({\mathbb{R}}_x^3))$ with $N\ge 3$. 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 $t\to \mathrm{\infty }$. Moreover, we derive the time-decay estimate of the perturbation around time-periodic solutions for initial data in $L^p$-norm with $3\mathrm{/}2<p\le 2$. 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.