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Production of the Fisher information for the Landau-Coulomb equation with L1 initial data

Laurent Desvillettes, William Golding, Maria Pia Gualdani, Amelie Loher

Abstract

We consider the Landau-Coulomb equation for initial data with bounded mass, finite numbers of moments, and entropy. We show the existence of a global weak solution that has bounded Fisher information for positive times. This solution is therefore a global strong solution away from the initial time. We propose an alternative approach, based on already existing estimates, to the study of the appearance of Fisher information recently performed by Ji in [12].

Production of the Fisher information for the Landau-Coulomb equation with L1 initial data

Abstract

We consider the Landau-Coulomb equation for initial data with bounded mass, finite numbers of moments, and entropy. We show the existence of a global weak solution that has bounded Fisher information for positive times. This solution is therefore a global strong solution away from the initial time. We propose an alternative approach, based on already existing estimates, to the study of the appearance of Fisher information recently performed by Ji in [12].

Paper Structure

This paper contains 6 sections, 9 theorems, 80 equations, 1 figure.

Table of Contents

  1. Proof of the main theorem
  2. Estimates on approximated coefficients
  3. The approximated solution: existence and properties
  4. Main uniform estimates
  5. The limit
  6. Coercivity for the truncated kernel

Key Result

Theorem 1

Let $f_{in}$ be a non-negative function such that The Landau-Coulomb equation with $f_{in}$ as initial datum has a global weak solution such that for any time $t\ge \varepsilon$ (for all $\varepsilon>0$ as small as one wishes), this weak solution is a strong solution and its Fisher information is monotone decreasing. More precisely, for any $t>0$, where $C'>0$ only depends on the mass, energy, e

Figures (1)

  • Figure 1: The timeline. Note that $t$ is independent of $n$, whereas $t_0$ and $t_1$ depend on $n$.

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 7 more