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

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.

arXiv comment21 pages, significant revisions to the uniqueness result in Theorem 1.3. The proof in the preceding version was incorrect and has been fixed