Weak-strong uniqueness for the Landau equation by a relative entropy method

TL;DR

This paper establishes a weak-strong uniqueness principle for the Landau equation with soft potentials, using a novel relative entropy approach to measure the distance between solutions, and provides conditions for initial data regularity.

Contribution

It introduces a new stability estimate for the Landau equation based on relative entropy, applicable to Coulomb interactions, and offers initial data regularity conditions.

Findings

Relative entropy effectively measures solution differences.

Weak-strong uniqueness holds under specified regularity.

Conditions for initial data ensure regularity in Coulomb case.

Abstract

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case.