Local Well-Posedness for Vlasov--Poisson with $L^{d+}$ Initial Density and Fractional Velocity Regularity

Abstract

We prove a local well-posedness criterion for the Vlasov--Poisson system on $\R^d_x\times\R^d_v$, $d\geq2$, under an anisotropic assumption on the initial distribution. The datum has finite mass, its weighted velocity supremum belongs to $L^p_x$ for some $p>d$, and it has an arbitrarily small positive H\"older regularity in the velocity variable, uniformly with respect to velocity and with the same spatial $L^p$ control. The main estimate is a nonlinear mixing bound \[ [\rho(t)]_{C^\alpha_x}\lesssim t^{-d/p-\epsilon}C(f_0), \qquad \epsilon>0~~\text{small}. \] Thus the density is integrable in time with values in a positive spatial H\"older class, and the corresponding electric field belongs to $L^1_tC^{1,\alpha}_x$. We construct a solution by a Schauder fixed point on the density and prove uniqueness by a Loeper-type stability estimate.