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

We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in}\in L^p$, where $p$ is arbitrarily close to $3\mathrm{/}2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p>3\mathrm{/}2$ 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 $L^p\to L^{\mathrm{\infty }}$ 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.