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Global solutions to the Landau-Fermi-Dirac equation

Paulo Sampaio

Abstract

We establish a compactness result for solutions of a certain class of hypoelliptic equations. This result allows us to show the existence of global weak solutions to the non-homogeneous Landau-Fermi-Dirac equation with Coulomb potential.

Global solutions to the Landau-Fermi-Dirac equation

Abstract

We establish a compactness result for solutions of a certain class of hypoelliptic equations. This result allows us to show the existence of global weak solutions to the non-homogeneous Landau-Fermi-Dirac equation with Coulomb potential.

Paper Structure

This paper contains 8 sections, 10 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. A compactness result for the LFD equation
  4. Velocity averages: compactness in time and position
  5. Quasi-ellipticity of the diffusion matrix: compactness in velocity
  6. Almost everywhere compactness
  7. Cauchy theory for the approximated equation
  8. The existence theorem

Key Result

Theorem 1

Let $N \geq 2$ and $f_0 \in L^1_2(\mathbb{R}^{2N}_{x,v})$ be such that $0 \leq f_0 \leq 1$. There exists a weak solution $f \in C((0,\infty);\mathcal{D}'(\mathbb{R}^{2N}_{x,v})) \cap L^\infty((0,\infty); L^1_2(\mathbb{R}^{2N}_{x,v}))$ to the LFD equation with initial data $f_0$, in the sense of Defi 2) Conservation of mass and linear momentum: 3) Decay of kinetic energy and moment of inertia: 4)

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Proposition 1
  • Theorem 2: Theorem 1.1.8 from bouchut-2000
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • ...and 11 more