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Dissipation estimates of the Fisher information for the Landau equation

Sehyun Ji

Abstract

We establish an a priori estimate for the dissipation of the Fisher information for the space-homogeneous Landau equation with very soft potentials. This work is motivated by the recent breakthrough by Guillen and Silvestre, which proves that the Fisher information is monotone decreasing. As a direct consequence, we show that the Fisher information becomes instantaneously bounded, even if it is not initially bounded. This leads to a proof of the global existence of smooth solutions for the space-homogeneous Landau equation with very soft potentials, given initial data $f_0 \in L^1_{2-γ} \cap L \log L$. This result includes the case of the Coulomb potential.

Dissipation estimates of the Fisher information for the Landau equation

Abstract

We establish an a priori estimate for the dissipation of the Fisher information for the space-homogeneous Landau equation with very soft potentials. This work is motivated by the recent breakthrough by Guillen and Silvestre, which proves that the Fisher information is monotone decreasing. As a direct consequence, we show that the Fisher information becomes instantaneously bounded, even if it is not initially bounded. This leads to a proof of the global existence of smooth solutions for the space-homogeneous Landau equation with very soft potentials, given initial data . This result includes the case of the Coulomb potential.

Paper Structure

This paper contains 9 sections, 18 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lifting
  4. The formula for the dissipation of the Fisher information
  5. Bakry-Émery $\Gamma_2$ criterion
  6. Known results
  7. A priori estimate for the dissipation of the Fisher information
  8. Global existence of smooth solutions for initial data in $L^1_{2-\gamma} \cap L\log L$
  9. A proof of Remark \ref{['remark: LlogL']}

Key Result

Theorem 1.2

Suppose $\alpha(r)=r^\gamma$ for $\gamma \in[-3,-2)$. Let $f:[0,T]\times \mathbb R^3$ be a solution of the Landau equation eq: Landaueq with initial data $f_0$ that has mass $M_0$, energy $E_0$, and entropy $H_0$. Then, Here $C_1,C_2$ are absolute constants and $c_1$ depends on $\gamma, M_0,E_0,H_0$.

Theorems & Definitions (32)

  • Definition 1.1: The Fisher information
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1: Lemma 3.4 and Remark 3.5 from guillen2023landau
  • Remark 2.2
  • Proposition 2.3: Lemma 8.2 from guillen2023landau
  • Theorem 2.4: Theorem 1.1 from ji2024bakryemery
  • ...and 22 more