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

TL;DR

The paper proves the uniqueness of H-solutions to the space-homogeneous Landau-Coulomb equation in the critical space L^{3/2}, under weighted regularity, and confirms the uniqueness of the GGL25-construction. The authors develop an M-operator framework with M as the Bessel potential to obtain a closed L^2 energy estimate for the difference of two solutions, enabling control by dissipation and commutator terms. Through detailed a priori estimates of multiple integral components and mollification arguments, they establish a robust uniqueness result for rough solutions and extend it to H-solutions with additional regularity. This work completes the global well-posedness theory in L^{3/2} and showcases the M-operator approach as a viable tool for nonlinear kinetic equations in rough spaces.

Abstract

We prove the uniqueness of $H$-solutions to the homogeneous Landau-Coulomb equation satisfying $\langle v \rangle^{k_0} f \in C([0, T]; L^{3/2}(\mathbb{R}^3))$ and $\langle v \rangle^{-3/2} \nabla_v ((\langle v \rangle^{k_0} f)^{3/4}) \in L^2((0, T) \times \mathbb{R}^3)$ for any $k_0 \geq 5$. 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.

Theorems & Definitions (32)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof