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 \(f_{in} \in L^p\) $f_{\mathit{in}}\in L^p$ , 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 \(p > 3/2\) $p>3/2$ 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 \(L^p \rightarrow L^\infty \) $L^p\to L^{\infty }$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large \(L^p\) $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying unweighted Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.