Schauder's estimate for nonlocal kinetic equations and its applications

Abstract

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in $\mathbb R^{2d}$ with Hölder coefficients: $$ \partial_tu=\mathscr L^{(α)}_{κ;{\rm v}} u+b\cdot\nabla u+f,\ u_0=0, $$ where $u=u(t,x,{\rm v})$ and $\mathscr L^{(α)}_{κ;{\rm v}}$ is a nonlocal $α$-stable-like operator with $α\in(1,2)$ and kernel function $κ$, which acts on the variable ${\rm v}$. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift $b$: $$ {\rm d}Z_t=b(t,Z_t){\rm d}t+(0,σ(t,Z_t){\rm d}L^{(α)}_t),\ \ Z_0=(x,{\rm v})\in\mathbb R^{2d}, $$ where $L^{(α)}_t$ is a $d$-dimensional rotationally invariant and symmetric $α$-stable process with $α\in(1,2)$, and $b:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^{2d}$ is a $(γ,β)$-Hölder continuous function in $(x,{\rm v})$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$ and $β\in\big(1-\fracα{2},1\big)$, $σ:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^d\otimes\mathbb R^d$ is a Lipschitz function. Moreover, we also show that for almost all $ω$, the following random transport equation has a unique $C^1_b$-solution: $$ \partial_tu(t,x,ω)+(b(t,x)+L^{(α)}_t(ω))\cdot\nabla_x u(t,x,ω)=0,\ \ u(0,x)=φ(x), $$ where $φ\in C^1_b(\mathbb R^d)$ and $b:\mathbb R_+\times\mathbb R^d\to\mathbb R^d$ is a bounded continuous function in $(t,x)$ and $γ$-order Hölder continuous in $x$ uniformly in $t$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$.