Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials

TL;DR

This paper proves well-posedness, stability, and regularity results for the homogeneous Landau equation with hard potentials, demonstrating instant creation of Gaussian moments and analytic densities from finite energy initial conditions.

Contribution

It extends regularity and stability results for the Landau equation, establishing instant Gaussian moment creation and analytic density regularity under minimal assumptions.

Findings

Proves well-posedness and stability using Monge-Kantorovich cost.

Shows solutions with finite energy develop analytic densities immediately.

Establishes existence of weak solutions with finite initial energy.

Abstract

We establish the well-posedness and some quantitative stability of the spatially homogeneous Landau equation for hard potentials, using some specific Monge-Kantorovich cost, assuming only that the initial condition is a probability measure with a finite moment of order $p$ for some $p>2$. As a consequence, we extend previous regularity results and show that all non-degenerate measure-valued solutions to the Landau equation, with a finite initial energy, immediately admit analytic densities with finite entropy. Along the way, we prove that the Landau equation instantaneously creates Gaussian moments. We also show existence of weak solutions under the only assumption of finite initial energy.