Vlasov-Poisson equation in weighted Sobolev space $W^{m, p}(w)$

Abstract

In this paper, we are concerned about the well-posedness of Vlasov-Poisson equation near vaccum in weighted Sobolev space $W^{m, p}(w).$ The most difficult part comes from estimates of the electronic term $\nabla_{x}\phi.$ To overcome this difficulty, we establish the $L^p$-$L^q$ estimates of the electronic term $\nabla_{x}\phi;$ some weight is introduced as well to obtain the off-diagonal estimate. The weight is also useful when it comes to control the higher-order derivative term.

Keywords

• vlasov-poisson
• lp-sobolev
• weighted estimates
• lp-lq estimates