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 Hs,p(w), that is, Bessel potential space. The difficulty lies in the estimates of the electronic term ∇xϕ. To overcome it, we establish the Lp-Lq estimate and take advantage of commutator estimates. The Lp-version energy method and pseudo-differential operator techniques are also introduced.