$L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators

Abstract

For $p\in (1,\infty )$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on ${\mathbb{R}}^{1+2d}$ in the weak sense: $${\partial }_{t}u+v\cdot {\nabla }_{x}u={\Delta }_v^{\alpha /2}u+f,$$ where $\alpha \in (0,2)$ and ${\Delta }_v^{\alpha /2}$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,\alpha ,d)>0$ such that for any $f(t,x,v)\in L^p(\mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^d)=L^p({\mathbb{R}}^{1+2d})$, $$\Vert {\Delta }_x^{\alpha /(2(1+\alpha ))}u{\Vert }_p+\Vert {\Delta }_v^{\alpha /2}u{\Vert }_p\le C\Vert f{\Vert }_p,$$ where $\Vert \cdot {\Vert }_p$ is the usual $L^p$-norm in $L^p({\mathbb{R}}^{1+2d};dz)$. In fact, in this paper the above inequality is established for a large class of time-dependent non-local kinetic Fokker-Plank equations on ${\mathbb{R}}^{1+2d}$, with $U_{t}v$ and ${\mathcal{L}}_{{\sigma }_t}^{{\nu }_t}$ in place of $v\cdot {\nabla }_x$ and ${\Delta }_v^{\alpha /2}$. See Theorem 3.3 for details.

36pages