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Global Existence for an Isotropic Landau Model

David Bowman, Sehyun Ji

Abstract

Following the recent ideas of Guillen and Silvestre in $[9]$, we prove that the Fisher information is non-increasing along the flow of the isotropic Landau equation. We then use this fact to deduce global existence for the equation $\partial_t f = (-Δ)^{-1}f \cdot Δf + f^2$ under a relatively lax set of conditions on the initial data. In particular, we remove the restrictive radially decreasing assumption of previous works.

Global Existence for an Isotropic Landau Model

Abstract

Following the recent ideas of Guillen and Silvestre in , we prove that the Fisher information is non-increasing along the flow of the isotropic Landau equation. We then use this fact to deduce global existence for the equation under a relatively lax set of conditions on the initial data. In particular, we remove the restrictive radially decreasing assumption of previous works.
Paper Structure (13 sections, 42 theorems, 257 equations)

This paper contains 13 sections, 42 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. History and Review of the Krieger-Strain equation
  3. Outline of Paper and Main Ideas
  4. Acknowledgements
  5. Preliminary Set-up
  6. The Maxwell Molecules Case
  7. The Fisher Information along the Krieger-Strain flow
  8. Propagation of Moments and an $L^\infty$ Estimate
  9. Propagation of $H^k$ Norms
  10. $L^\infty$ Local Existence Theory
  11. Concluding the Global Existence Theorem
  12. Technical Inequalities
  13. Proof of Proposition \ref{['prop: localexistenceHK']}: $H^3_q$ Local Existence Theory

Key Result

Theorem 1.1

Let $f:[0,T] \times \mathbb R^3 \to [0,\infty)$ be a classical solution with rapid decay to the equation where $\alpha:[0,\infty) \to [0,\infty)$ is an arbitrary interaction potential. Assume that $\alpha(\cdot)$ satisfies Then the Fisher information $i(f)$ is monotone decreasing as a function of time. In particular, taking $\alpha(r) = r^\gamma$, we recover the monotonicity of the Fisher inform

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: Essentially from Lemma $4.1$ of guillen2023global
  • Lemma 2.6
  • proof
  • ...and 59 more