Vlasov–Poisson Equation in Hs,p(w) Space

Abstract

In this paper, we are concerned with the well-posedness of Vlasov–Poisson equation near vacuum in weighted fractional Sobolev space ${\mathit{H}}^{\mathit{s},\mathit{p}}(\mathit{w})$, that is, Bessel potential space. The difficulty lies in the estimates of the electronic term ${\nabla }_{\mathit{x}}\mathit{\varphi }$. To overcome it, we establish the ${\mathit{L}}^{\mathit{p}}$-${\mathit{L}}^{\mathit{q}}$ estimate and take advantage of commutator estimates. The ${\mathit{L}}^{\mathit{p}}$-version energy method and pseudo-differential operator techniques are also introduced.