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Global smooth solutions to the Landau-Coulomb equation in $L^{3/2}$

William Golding, Maria Gualdani, Amélie Loher

TL;DR

The paper proves global-in-time smooth solutions to the homogeneous Landau equation with Coulomb potential for initial data in the critical weighted space $L^{3/2}$ and extends well-posedness to $L^p$-based data with $p>3/2$. It introduces a new $oldsymbol{}$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg framework, linking smallness in a weighted $L^{3/2}$ norm to regularity, and leverages short-time regularization of the Fisher information $i(f)$ to bootstrap to global solutions. The analysis handles the nonlocal, unbounded diffusion coefficients by establishing precise weighted estimates, De Giorgi-type regularization, and a two-track local/global theory (critical $p=3/2$ and subcritical $p>3/2$), culminating in global well-posedness with decreasing Fisher information and, in favorable cases, convergence toward Maxwellian equilibrium. The results significantly advance the understanding of global dynamics for rough, slowly decaying initial data in the Landau-Coulomb setting and provide a robust framework for further study of inhomogeneous models. The methods blend De Giorgi, energy methods, weighted Sobolev inequalities, and monotonicity of the Fisher information to overcome the challenges posed by the critical $L^{3/2}$ scale and nonuniform ellipticity.

Abstract

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to $L^p$ with $p>3/2$, there is a unique solution. At the crux of the result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the $L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.

Global smooth solutions to the Landau-Coulomb equation in $L^{3/2}$

TL;DR

The paper proves global-in-time smooth solutions to the homogeneous Landau equation with Coulomb potential for initial data in the critical weighted space and extends well-posedness to -based data with . It introduces a new -regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg framework, linking smallness in a weighted norm to regularity, and leverages short-time regularization of the Fisher information to bootstrap to global solutions. The analysis handles the nonlocal, unbounded diffusion coefficients by establishing precise weighted estimates, De Giorgi-type regularization, and a two-track local/global theory (critical and subcritical ), culminating in global well-posedness with decreasing Fisher information and, in favorable cases, convergence toward Maxwellian equilibrium. The results significantly advance the understanding of global dynamics for rough, slowly decaying initial data in the Landau-Coulomb setting and provide a robust framework for further study of inhomogeneous models. The methods blend De Giorgi, energy methods, weighted Sobolev inequalities, and monotonicity of the Fisher information to overcome the challenges posed by the critical scale and nonuniform ellipticity.

Abstract

We consider the homogeneous Landau equation in with Coulomb potential and initial data in polynomially weighted . We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to with , there is a unique solution. At the crux of the result is a new -regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted is regular. Although the norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the norm alone is not enough to control the norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.
Paper Structure (20 sections, 29 theorems, 257 equations)

This paper contains 20 sections, 29 theorems, 257 equations.

Table of Contents

  1. Known results and technical lemmas
  2. Prior well-posedness results and proof of Theorem \ref{['thm:long-cor-GS-GL']}
  3. An epsilon-regularity criterion for Landau-Coulomb
  4. The local in time theory
  5. Propagation of for smooth data
  6. Construction of solution
  7. The global in time theory
  8. Pointwise lower bounds
  9. Bound on the Fisher information and proof of Theorem \ref{['thm:long']}
  10. The subcritical local in time theory
  11. Subcritical De Giorgi
  12. Propagation of Lpm for smooth data
  13. Construction of solution
  14. Uniqueness
  15. The subcritical global in time theory
  16. ...and 5 more sections

Key Result

Theorem 1

(Local-in-Time well-posedness) Let $\frac{3}{2}\le p < +\infty$. If the initial data $f_{in}$ is such that $f_{in}\langle v \rangle^{\frac{9}{2p}} \in L^p(\mathbb{R}^3)$, then there exists a smooth solution $f:[0,T]\times {\mathbb R}^3 \to {\mathbb R}^+$ to (eq:landau) such that Moreover, if $p > 3/2$, this solution is unique among all smooth solutionsMore precisely, following GoldingLoher, we

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Corollary 5
  • Lemma 1.1: Propagation of moments CarrapatosoDesvillettesHe
  • Theorem 1.2: Uniqueness Fournier
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • ...and 33 more