Well-posedness for Vlasov--Poisson on Low Regularity $H^{s,p}$ Spaces in All Dimensions

We consider the Vlasov–Poisson equation on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ for any dimension. For initial distribution $f_0$ having compact support in $v$ and belonging to $H^{s,p}({\mathbb{R}}^d\times {\mathbb{R}}^d)$, we prove local well-posedness for $s>\frac{d}{p}-\frac{1}{2p}$ and $p\in [2,\mathrm{\infty })$. We proved this by using two main ingredients, which is the averaging property for the density $\rho$ and the Schauder-Tychonoff fixed point theorem. It seems like this is the first application of the Schauder-Tychonoff fixed point theorem in showing existence of solution for evolutionary PDEs.