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The Landau equation does not blow up

Nestor Guillen, Luis Silvestre

TL;DR

This work proves that the Fisher information for the space-homogeneous Landau equation with a broad class of interaction potentials is monotone decreasing in time under a natural one-parameter threshold, thereby ruling out finite-time blow-up and yielding global smooth solutions in the very-soft-potential range (including Coulomb). The key technique is lifting the nonlinear, nonlocal Landau operator to a linear degenerate parabolic operator on $\mathbb{R}^6$, $Q$, acting on $F=f\otimes f$, and showing $\langle I'(F),Q(F)\rangle\le0$ by decomposing $Q$ into directional diffusions along three lifted vector fields and exploiting a sharp inequality on the sphere. A crucial step is a new Poincaré-type inequality on $S^2$ with constant $\tfrac{19}{4}$, enabling control of the remainder terms and compatibility with the level-set geometry of $|v-w|$. The results yield global existence of smooth solutions for $\alpha(r)=r^\gamma$ with $\gamma\in[-3,1]$, including the Coulomb case $\gamma=-3$, thereby advancing long-standing regularity questions for the Landau equation in kinetic theory.

Abstract

We consider solutions to the space-homogeneous Landau equation with a general family of interaction potentials. We prove that their Fisher information is monotone decreasing in time. The class of interaction potentials covered by our result includes the case of the Landau equation with Coulomb interactions. As a consequence of the global boundedness of the Fisher information, we deduce that solutions to the space-homogeneous Landau equation never blow up.

The Landau equation does not blow up

TL;DR

This work proves that the Fisher information for the space-homogeneous Landau equation with a broad class of interaction potentials is monotone decreasing in time under a natural one-parameter threshold, thereby ruling out finite-time blow-up and yielding global smooth solutions in the very-soft-potential range (including Coulomb). The key technique is lifting the nonlinear, nonlocal Landau operator to a linear degenerate parabolic operator on , , acting on , and showing by decomposing into directional diffusions along three lifted vector fields and exploiting a sharp inequality on the sphere. A crucial step is a new Poincaré-type inequality on with constant , enabling control of the remainder terms and compatibility with the level-set geometry of . The results yield global existence of smooth solutions for with , including the Coulomb case , thereby advancing long-standing regularity questions for the Landau equation in kinetic theory.

Abstract

We consider solutions to the space-homogeneous Landau equation with a general family of interaction potentials. We prove that their Fisher information is monotone decreasing in time. The class of interaction potentials covered by our result includes the case of the Landau equation with Coulomb interactions. As a consequence of the global boundedness of the Fisher information, we deduce that solutions to the space-homogeneous Landau equation never blow up.
Paper Structure (15 sections, 41 theorems, 208 equations)

This paper contains 15 sections, 41 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Historical remarks and references
  3. Outline of the paper and main ideas in the proof.
  4. Preliminaries
  5. Lifting and projection
  6. Flowing along vector fields
  7. The case of Maxwell molecules
  8. The Fisher information along layers
  9. Using commutators to estimate the remaining directional derivative
  10. The three distinct diffusion terms
  11. An inequality for functions on the sphere
  12. The monotonicity of the Fisher information
  13. The global existence theorem
  14. the two-dimensional case
  15. On the decay of the tails of our integrals

Key Result

Theorem 1.1

Let $f : [0,T] \times \mathbb R^3 \to [0,\infty)$ be a classical solution to the space-homogeneous Landau equation e:landauequation. Assume that the interaction potential $\alpha$ satisfies, for all $r>0$, then the Fisher information $i(f)$ is monotone decreasing as a function of time.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1: from henderson2020
  • Theorem 2.2: from silvestre2017
  • Theorem 2.3: essentially from cameron2018
  • proof : Sketch proof
  • Theorem 2.4
  • Theorem 2.5
  • ...and 77 more