Vlasov–Poisson–Fokker–Planck equation in E2,1s space

Abstract

In this paper, we are concerned about the local-in-time well-posedness of the Vlasov–Poisson–Fokker–Planck equation in $E_{2,1}^s({\mathbb{R}}^{2n})$ which is a hybrid modulation Lebesgue space and related to the Gevery class only with respect to x variable. The difficulty lies in the estimates of the electronic term ${\nabla }_x\mathit{\varphi }$. To handle this, we establish a product formula and $L^2$-$L^q$ estimate.