Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation

Abstract

We prove a Schauder estimate for kinetic Fokker-Planck equations that requires only H\"older regularity in space and velocity but not in time. As an application, we deduce a weak-strong uniqueness result of classical solutions to the spatially inhomogeneous Landau equation beginning from initial data having H\"older regularity in $x$ and only a logarithmic modulus of continuity in $v$. This replaces an earlier result requiring H\"older continuity in both variables.