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

William Golding, Maria Gualdani, Amélie Loher

Published 2024-01-13Version 1

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 crux of this result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem and short-time regularization estimates for the Fisher information. The resulting estimate, 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. Complications arise when $p = 3/2$ as a result of the loss of regularity and boundedness in the coefficients. This introduces challenges that mark the threshold where prior methods break down. Our framework is general enough to handle slowly decaying and potentially singular initial data and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.