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The Landau equation as a Gradient Flow

José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes, Jeremy S. H. Wu

Abstract

We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the Energy Dissipation Inequality. Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.

The Landau equation as a Gradient Flow

Abstract

We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the Energy Dissipation Inequality. Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.

Paper Structure

This paper contains 18 sections, 34 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main results
  3. Notations and definitions
  4. Quick review of gradient flow theory
  5. Main results
  6. The Landau metric $d_L$
  7. Grazing continuity equation
  8. Action of a curve
  9. Properties of the Landau metric
  10. Energy dissipation equality
  11. JKO scheme for $\epsilon$-Landau equation
  12. Recovering the full Landau equation as $\epsilon \to 0$
  13. Outline of technical strategy to prove \ref{['eq:tedct']}
  14. Preparatory results
  15. Proof of \ref{['eq:tedct']} using EDCT \ref{['thm:EDCT']}
  16. ...and 3 more sections

Key Result

Theorem \oldthetheorem

The (pseudo)-metric $d_L$ on $\mathscr{P}_{2,E}(\mathbb{R}^d)$ satisfies:

Theorems & Definitions (85)

  • Definition \oldthetheorem: Weak $\mathcal{F}$ solutions
  • Definition \oldthetheorem: Absolutely continuous curve
  • Definition \oldthetheorem: Metric derivative
  • Definition \oldthetheorem: Strong upper gradient
  • Definition \oldthetheorem: Curve of maximal slope
  • Definition \oldthetheorem: Slopes
  • Theorem \oldthetheorem: Distance on $\mathscr{P}_{2,E}(\mathbb{R}^d)$
  • Theorem \oldthetheorem: Epsilon equivalence
  • Theorem \oldthetheorem: Existence of curves of maximal slope
  • Remark \oldthetheorem
  • ...and 75 more