Local-In-Time Strong Solutions of the Homogeneous Landau–Coulomb Equation with Lp Initial Datum

We consider the homogeneous Landau equation with Coulomb potential and general initial data f 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 → L ∞ 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 unweighted Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.