Uniqueness for the Homogeneous Landau-Coulomb Equation in $L^{3/2}$

Abstract

We prove the uniqueness of $H$-solutions to the homogeneous Landau-Coulomb equation satisfying $⟨v{⟩}^{k_0}f\in C([0,T];L^{3/2}({\mathbb{R}}^3))$ and $⟨v{⟩}^{-3/2}{\nabla }_v((⟨v{⟩}^{k_0}f{)}^{3/4})\in L^2((0,T)\times {\mathbb{R}}^3)$ for any $k_0\ge 5$. In particular, this shows that the solutions constructed in~\cite{GGL25} are unique. The present work thus completes the global well-posedness theory in the critical space $L^{3/2}({\mathbb{R}}^3)$. Our proof is part of a broader effort to use the $\mathcal{M}$-operator technique developed in~\cite{AGS2025, AMSY2020} to establish the uniqueness of rough solutions to nonlinear kinetic equations. When applied to the space-homogeneous case, the $\mathbb{M}$-operator can be taken simply as a Bessel potential operator.