Local existence for the Landau Equation with hard potentials

TL;DR

This paper establishes local existence and uniqueness of solutions for the inhomogeneous Landau equation with hard potentials on the whole space, overcoming previous challenges related to moment loss by introducing a novel weighted norm hierarchy.

Contribution

It provides the first local existence and uniqueness results for the Landau equation with hard potentials, using a new weighted hierarchy of norms to handle moment loss issues.

Findings

Proved local existence and uniqueness for hard potentials.

Developed a weighted hierarchy of norms for the analysis.

Addressed the moment loss problem specific to hard potentials.

Abstract

We consider the spatially inhomogeneous Landau equation with hard potentials (i.e. with $\gamma \in [0,1]$) on the whole space $\mathbb{R}^3$. We prove existence and uniquenss of solutions for a small time, assuming that the initial data is in a weighted tenth-order Sobolev space and has exponential decay in the velocity variable. In constrast to the soft potential case, local existence for the hard potentials case has been missing from the literature. This is because the moment loss issue is the most severe for these potentials. To get over this issue, our proof relies on a weighted hierarchy of norms that depends on the number of spatial and velocity derivatives in an asymmetric way. This hierarchy lets us take care of the terms that are affected by the moment loss issue the most. These terms do not give in to methods applied to study existence of solutions to Landau equation with soft potentials and are a major reason why the local existence problem was not known for the case of hard potentials.