KINETIC SCHAUDER ESTIMATES WITH TIME-IRREGULAR COEFFICIENTS AND UNIQUENESS FOR THE LANDAU EQUATION

We prove a Schauder estimate for kinetic Fokker-Planck equations that requires only Hölder regularity in space and velocity but not in time; we require only measurability in time. This allows us to sidestep a major technical issue for kinetic equations by decoupling the time, space and velocity variables, which are intertwined by the transport operator. As an application, we consider the spatially inhomogeneous Landau equation. Leveraging the convolutional nature of the coefficients (which yields extra v-regularity) and applying our new estimates, we deduce a weak-strong uniqueness result of classical solutions beginning from initial data having Hölder regularity in x and only a logarithmic modulus of continuity in v. This replaces an earlier result requiring Hölder continuity in both variables and indicates that well-posedness requires less regularity than previously thought.