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Some remarks about the Link between the Fisher information and Landau or Landau-Fermi-Dirac entropy dissipation

Laurent Desvillettes

Abstract

We present in this work variants of existing estimates for the Landau or Landau-Fermi-Dirac entropy dissipation, in terms of Fisher information, in the hard potential case. The specificity of those variants is that the entropy is never used in the estimates (in order to control possible concentrations on a zero measure set). The proofs are significantly simplified with respect to previous papers on the subject

Some remarks about the Link between the Fisher information and Landau or Landau-Fermi-Dirac entropy dissipation

Abstract

We present in this work variants of existing estimates for the Landau or Landau-Fermi-Dirac entropy dissipation, in terms of Fisher information, in the hard potential case. The specificity of those variants is that the entropy is never used in the estimates (in order to control possible concentrations on a zero measure set). The proofs are significantly simplified with respect to previous papers on the subject
Paper Structure (5 sections, 3 theorems, 112 equations)

This paper contains 5 sections, 3 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Description of the Landau operator and its entropy dissipation
  3. Proof of Theorem \ref{['main']}
  4. Application to the large time behavior of the Landau equation
  5. Extension to the Landau-Fermi-Dirac equation

Key Result

Theorem 1

We consider $\gamma$ satisfying (hshp). Then for all $f:= f(v) > 0$ (sufficiently smooth for the terms in the estimate to make sense) satisfying and such that the following estimate holds:

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Remark 2
  • Proposition 2
  • Remark 3