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

TL;DR

This paper establishes a Schauder estimate for kinetic Fokker-Planck equations with minimal regularity assumptions and applies it to prove weak-strong uniqueness for the Landau equation starting from less regular initial data.

Contribution

It introduces a novel Schauder estimate that requires only Hölder regularity in space and velocity, not in time, and applies it to prove uniqueness for the Landau equation with weaker initial regularity.

Findings

Schauder estimate for kinetic equations with time-irregular coefficients

Weak-strong uniqueness for the Landau equation with minimal initial regularity

Reduction of regularity assumptions compared to previous results

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.