Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data

TL;DR

This paper establishes the existence and uniqueness of solutions to the inhomogeneous Landau equation with rough, large initial data, including vacuum regions, under weaker regularity and decay assumptions than previous results.

Contribution

It introduces a new framework for solving the Landau equation with minimal regularity and decay requirements, extending the class of initial data for which solutions are known to exist and be unique.

Findings

Constructed solutions for large, rough initial data with polynomial decay

Proved uniqueness under H"older continuity of initial data

Derived an improved continuation criterion for very soft potentials

Abstract

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is H\"older continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art.