Low regularity Sobolev well-posedness for Vlasov--Poisson

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.

17 pages