Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum

TL;DR

This paper proves local-in-time existence and uniqueness of smooth solutions to the homogeneous Landau-Coulomb equation with initial data in $L^p$, near $p=3/2$, using new smoothing estimates and regularity techniques.

Contribution

It introduces new short-time $L^p$ to $L^ty$ smoothing estimates and extends regularity and uniqueness results for solutions with minimal integrability assumptions.

Findings

Established local existence and uniqueness for initial data in $L^p$ with $p$ close to 3/2.

Derived new short-time smoothing estimates from $L^p$ to $L^ty$.

Proved conditional regularity and stability results under unweighted Prodi-Serrin conditions.

Abstract

We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.