Global smooth solutions to the Landau-Coulomb equation in $L^{3/2}$

TL;DR

This paper proves the existence of global smooth solutions to the Landau-Coulomb equation in three dimensions with initial data in a critical $L^{3/2}$ space, overcoming significant analytical challenges.

Contribution

It introduces a new $ extit{ extbf{ε}}$-regularity criterion and a novel framework for handling rough initial data in the critical $L^{3/2}$ space for the Landau-Coulomb equation.

Findings

Existence of global smooth solutions for initial data in $L^{3/2}$.

Uniqueness of solutions when initial data is in $L^p$ with $p>3/2$.

First proof of global well-posedness with rough initial data.

Abstract

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to $L^p$ with $p>3/2$, there is a unique solution. At the crux of the result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the $L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.