Low regularity Sobolev well-posedness for Vlasov--Poisson

TL;DR

The work advances the theory of the Vlasov--Poisson equation by establishing local well-posedness in $H^{s}$ for $s> n/2-1/4$ on $\mathbb{R}^n\times\mathbb{R}^n$ ($n\ge 3$) with initial data compactly supported in velocity. It combines sharp a priori estimates, velocity averaging to control the potential, and a constructive regularization scheme to obtain a distributional solution that persists in $C([0,T];H^{s})$, along with a Loeper-type uniqueness argument based on transport along the characteristic flow. A key novelty is the low regularity threshold, made possible by the velocity-averaging gain of $1/4$ derivative, which allows data that are not necessarily in high $L^{p}$ spaces. Overall, the results broaden the admissible initial data class and illuminate the role of averaging effects in the dynamics of collisionless plasmas and related kinetic systems.

Abstract

We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ having compact support in $v$. In particular, data not belonging to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for large $p$ are allowed.

Theorems & Definitions (7)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• Lemma 2.2: AgoGLPS,Glassey96
• Lemma 2.3
• Lemma 4.1
• proof