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Global weak solutions for a nonlocal multispecies Fokker-Planck-Landau system

Jingwei Hu, Ansgar Jüngel, Nicola Zamponi

Abstract

The global-in-time existence of weak solutions to a spatially homogeneous multispecies Fokker-Planck-Landau system for plasmas in the three-dimensional whole space is shown. The Fokker-Planck-Landau system is a simplification of the Landau equations assuming a linearized, velocity-independent, and isotropic kernel. The resulting equations depend nonlocally and nonlinearly on the moments of the distribution functions via the multispecies local Maxwellians. The existence proof is based on a three-level approximation scheme, energy and entropy estimates, as well as compactness results, and it holds for both soft and hard potentials.

Global weak solutions for a nonlocal multispecies Fokker-Planck-Landau system

Abstract

The global-in-time existence of weak solutions to a spatially homogeneous multispecies Fokker-Planck-Landau system for plasmas in the three-dimensional whole space is shown. The Fokker-Planck-Landau system is a simplification of the Landau equations assuming a linearized, velocity-independent, and isotropic kernel. The resulting equations depend nonlocally and nonlinearly on the moments of the distribution functions via the multispecies local Maxwellians. The existence proof is based on a three-level approximation scheme, energy and entropy estimates, as well as compactness results, and it holds for both soft and hard potentials.
Paper Structure (11 sections, 11 theorems, 153 equations)

This paper contains 11 sections, 11 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Motivation of the model and some properties
  3. The homogeneous Fokker--Planck--Landau system
  4. The homogeneous linearized Fokker--Planck--Landau system
  5. Proof of Theorem \ref{['thm.ex']}
  6. Existence of solutions to the approximated system
  7. Limit $M\to\infty$
  8. Limit $\varepsilon\to 0$
  9. Limit $\delta\to 0$
  10. A compactness result
  11. Rigorous test functions

Key Result

Theorem 1

Let $f_i^0\in L^1({\mathbb R}^3;\langle v\rangle^2\mathrm{d}v)$ be nonnegative with $\int_{{\mathbb R}^3}f_i^0\log f_i^0\mathrm{d}v<\infty$, let $\gamma\in{\mathbb R}$, and let the constants $m_i,q_i,\Lambda,\varepsilon_0>0$ for $i=1,\ldots,s$. Then, for any $T>0$, there exists a nonnegative weak so Moreover, there exists a constant $c>0$ such that $T_{ji}(t)\ge c>0$, $c_{ji}(t)\ge c>0$ for $t\in(

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 2: Conservation properties
  • Lemma 3: Entropy decay
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 12 more