Existence of smooth solutions to the Landau equation with hard potentials and irregular initial data

TL;DR

This paper proves the existence of smooth solutions to the Landau equation with hard potentials from irregular initial data, extending previous results by relaxing initial data conditions and establishing regularity and uniqueness.

Contribution

It improves existing results by showing solutions become infinitely smooth for positive times and allows less restrictive initial data in sub-exponentially-weighted spaces.

Findings

Solutions are $C^ abla$ for positive times.

Existence holds for initial data in sub-exponentially-weighted $L^\infty$ spaces.

Uniqueness requires H"older continuity and no vacuum regions.

Abstract

This paper addresses large-data local existence and uniqueness of classical solutions to the inhomogeneous Landau equation in the hard potentials case (including Maxwell molecules). Solutions have previously been constructed by Chaturvedi [SIAM J. Math. Anal., 55(5), 5345--5385, 2023] for initial data in an exponentially-weighted $H^{10}$ space, but it is not a priori clear whether these solutions have more regularity than the initial data. We improve Chaturvedi's existence result in two ways: our solutions are $C^\infty$ for positive times, and we allow initial data in a sub-exponentially-weighted $L^\infty$ space, at the cost of requiring a mild positivity condition at time zero. To prove uniqueness, we require stronger assumptions on the initial data: H\"older continuity and the absence of vacuum regions. These are the same assumptions that are required for uniqueness in prior work on the soft potentials case. Along the way to proving existence and uniqueness, we establish some useful results that were previously only known in the case of soft potentials, including spreading of positivity and propagation of H\"older continuity. Many of the proof strategies from the soft potentials case do not apply here because of the more severe loss of velocity moments.