Abstract: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 $p\in[2, \infty)$. 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.
From: Sangwook Tae [view email]
[v1]
Sat, 13 Jun 2026 12:41:09 UTC (35 KB)
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