[Paper Review] $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators

Zhen-Qing Chen, Xicheng Zhang|arXiv (Cornell University)|Aug 19, 2016

Advanced Mathematical Physics Problems7 references3 citations

TL;DR

This paper establishes $L^p$-maximal hypoelliptic regularity for a class of nonlocal kinetic Fokker-Planck equations with time-dependent coefficients, proving that solutions gain fractional spatial and velocity regularity proportional to the $L^p$-norm of the forcing term. The key result is a sharp estimate: $\|\Delta_{x}^{\alpha/(2(1+\alpha))}u\|_p + \|\Delta_{\mathrm{v}}^{\alpha/2}u\|_p \leq C\|f\|_p$ for $p \in (1,\infty)$, extending classical hypoelliptic theory to nonlocal operators.

ABSTRACT

For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} imes \mathbb{R}^d imes \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t u+v\cdot abla_x u=Δ^{α/{2}}_v u+f, $$ where $α\in(0,2)$ and $Δ^{α/{2}}_v$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,α,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} imes \mathbb{R}^d imes \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$ \|Δ_x^{α/{(2(1+α))}}u\|_p+\|Δ_v^{α/{2}}u\|_p\leq C\|f\|_p, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb{R}^{1+2d}; d z)$. 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 $\mathscr{L}^{ν_t}_{σ_t}$ in place of $v\cdot abla_x$ and $Δ^{α/2}_v$. See Theorem 3.3 for details.

Motivation & Objective

• To extend $L^p$-regularity theory from local to nonlocal kinetic Fokker-Planck operators with fractional Laplacian in velocity.
• To establish sharp fractional smoothing estimates for solutions in both position and velocity variables.
• To generalize known $L^2$-results to $L^p$-spaces for $p \in (1,\infty)$ using Fourier analysis and singular integral techniques.
• To address the hypoelliptic regularity of time-dependent nonlocal kinetic equations with general time-dependent coefficients $U_t$ and $\mathscr{L}^{\nu_t}_{\sigma_t}$.

Proposed method

• Use of Fourier transformation to analyze the operator in frequency space, transforming the PDE into a multiplier problem.
• Application of Hölder’s inequality and Fubini’s theorem to control mixed norms in time, position, and velocity variables.
• Estimation of oscillatory integrals via exponential decay bounds derived from the symbol $\psi^{\nu_r}_{\sigma_r}$ of the nonlocal operator.
• Use of change of variables and symmetric estimates to bound the $L^2$-norm of the fractional Laplacian of the solution.
• Establishment of weak-type $(1,1)$ estimates and interpolation to extend $L^2$ results to $L^p$ for $p \in (1,\infty)$.
• Proof of the main inequality via a limiting procedure and approximation by smooth compactly supported functions.

Experimental results

Research questions

• RQ1 Does the fractional hypoelliptic regularity estimate $\|\Delta_x^{\alpha/(2(1+\alpha))}u\|_p + \|\Delta_{\mathrm{v}}^{\alpha/2}u\|_p \leq C\|f\|_p$ hold for $p \in (1,\infty)$, extending known $L^2$ results?
• RQ2 Can the $L^p$-maximal hypoelliptic regularity be established for time-dependent nonlocal kinetic Fokker-Planck operators with general time-dependent coefficients?
• RQ3 What is the precise scaling of regularity gain in position and velocity variables for solutions to nonlocal kinetic equations with fractional Laplacian in velocity?
• RQ4 How does the interplay between transport and nonlocal diffusion affect the $L^p$-regularity of solutions?

Key findings

• The paper establishes the sharp $L^p$-maximal hypoelliptic regularity estimate: $\|\Delta_x^{\alpha/(2(1+\alpha))}u\|_p + \|\Delta_{\mathrm{v}}^{\alpha/2}u\|_p \leq C\|f\|_p$ for all $p \in (1,\infty)$, with $C = C(p,\alpha,d) > 0$.
• The result holds for a large class of time-dependent nonlocal kinetic Fokker-Planck equations with $U_t{\mathrm{v}}$ and $\mathscr{L}^{\nu_t}_{\sigma_t}$ replacing ${{\mathrm{v}}\cdot\nabla_x}$ and $\Delta^{\alpha/2}_{\mathrm{v}}$.
• For $p=2$, the proof relies on Fourier analysis, $L^2$-boundedness of oscillatory integrals, and exponential decay estimates of the symbol $\psi^{\nu_r}_{\sigma_r}$.
• The constant $C$ in the estimate is independent of the solution and depends only on $p$, $\alpha$, and the dimension $d$.
• The method extends previous $L^2$ results by Alexander (2015) and Bouchut (2002) to the full range $p \in (1,\infty)$ via interpolation and weak-type estimates.
• The scaling $\alpha/(2(1+\alpha))$ in the position variable is optimal and matches the known $L^2$-case, confirming the fractional hypoelliptic nature of the operator.