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

2025/12/24 by Maria Pia Gualdani, Gualdani, Maria Pia, Weiran Sun +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2512.20899

openalex publication_date 2025/12/24 · openalex created_date 2025/12/26 · openalex updated_date 2026/07/28

Abstract

We prove the uniqueness of H-solutions to the homogeneous Landau-Coulomb equation satisfying ⟨ v ⟩^k₀ f ∈ C([0, T]; L^3/2(ℝ³)) and ⟨ v ⟩^-3/2 ∇_v ((⟨ v ⟩^k₀ f)^3/4) ∈ L²((0, T) × ℝ³) for any k₀ ≥ 5. In particular, this shows that the solutions constructed in~\citeGGL25 are unique. The present work thus completes the global well-posedness theory in the critical space L^3/2(ℝ³). Our proof is part of a broader effort to use the M-operator technique developed in~\citeAGS2025, AMSY2020 to establish the uniqueness of rough solutions to nonlinear kinetic equations. When applied to the space-homogeneous case, the \mathbbM-operator can be taken simply as a Bessel potential operator.

Citations

• Weak-strong uniqueness for the Landau equation by a relative entropy method
• Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in H^-\f12 space with polynomial tail
• Dissipation estimates of the Fisher information for the Landau equation
• Global smooth solutions to the Landau-Coulomb equation in L^3/2
• The Landau equation does not blow up
• Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with L^p initial datum
• Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation
• Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data
• Non-Cutoff Boltzmann Equation with Polynomial Decay Perturbation
• Nonuniqueness of weak solutions to the Navier-Stokes equation
• Upper bounds for parabolic equations and the Landau equation
• Entropy dissipation estimates for the Landau equation in the Coulomb\n case and applications

Related

• Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions
• Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in H^-\f12 space with polynomial tail
• Global well-posedness for the homogeneous Landau equation
• Global smooth solutions to the Landau-Coulomb equation in L^3/2
• Global in Time Estimates for the Spatially Homogeneous Landau Equation with Soft Potentials