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On the existence of solutions to the multi-species Landau equation

Jonathan Junné, Raphael Winter, Havva Yoldaş

TL;DR

This work extends the Fisher-information-based global well-posedness framework from the single-species Landau equation to the spatially homogeneous multi-species setting, identifying a sharp threshold γ ≥ -\\sqrt{8} for cross-interaction dissipation and establishing a global solution theory for finite-mass mixtures. When γ falls below the threshold, Fisher information ceases to be a universal Lyapunov functional; however, a novel Lyapunov functional combining I, a spherical Fisher term J, and entropy ensures global well-posedness in the physically relevant infinite-mass two-species limit (electrons-ion interactions) for γ ∈ [ -3, -\\sqrt{8} ). The paper also develops a lifting framework to relate multi-species dynamics to a lifted system, along with rigorous local well-posedness and moment control, thereby providing a comprehensive approach to global existence for soft potentials in multi-species kinetic theory.

Abstract

We consider the spatially homogeneous Landau equation for multiple species with different masses. As in the single-species case, the singularity of the collision operator is determined by a parameter $γ\in [-3,1]$, where $γ= -3$ corresponds to Coulomb interactions. We prove that if $γ\geq -\sqrt{8}$ in the cross-interaction operators, then there exists a natural multi-species generalization of the Fisher information which is a Lyapunov functional for the multi-species Landau system. On the other hand, we give a counterexample showing that the Fisher information is in general no longer a Lyapunov functional below the threshold $(γ< - \sqrt{8})$ for the two-species system if one species has infinite mass. However, we are able to provide a new method to show global well-posedness, by constructing a different Lyapunov functional based on the spherical Fisher information.

On the existence of solutions to the multi-species Landau equation

TL;DR

This work extends the Fisher-information-based global well-posedness framework from the single-species Landau equation to the spatially homogeneous multi-species setting, identifying a sharp threshold γ ≥ -\\sqrt{8} for cross-interaction dissipation and establishing a global solution theory for finite-mass mixtures. When γ falls below the threshold, Fisher information ceases to be a universal Lyapunov functional; however, a novel Lyapunov functional combining I, a spherical Fisher term J, and entropy ensures global well-posedness in the physically relevant infinite-mass two-species limit (electrons-ion interactions) for γ ∈ [ -3, -\\sqrt{8} ). The paper also develops a lifting framework to relate multi-species dynamics to a lifted system, along with rigorous local well-posedness and moment control, thereby providing a comprehensive approach to global existence for soft potentials in multi-species kinetic theory.

Abstract

We consider the spatially homogeneous Landau equation for multiple species with different masses. As in the single-species case, the singularity of the collision operator is determined by a parameter , where corresponds to Coulomb interactions. We prove that if in the cross-interaction operators, then there exists a natural multi-species generalization of the Fisher information which is a Lyapunov functional for the multi-species Landau system. On the other hand, we give a counterexample showing that the Fisher information is in general no longer a Lyapunov functional below the threshold for the two-species system if one species has infinite mass. However, we are able to provide a new method to show global well-posedness, by constructing a different Lyapunov functional based on the spherical Fisher information.

Paper Structure

This paper contains 14 sections, 19 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Summary of previous results
  3. Main results
  4. Preliminaries and structure of the paper
  5. The multi-species Landau system with finite masses
  6. Lifting argument for the multi-species system
  7. Dissipation of the weighted Fisher information
  8. Local well-posedness
  9. Global well-posedness
  10. The two-species Landau system with one species having infinite mass
  11. Dissipation identities and spherical Fisher information
  12. Non-monotonicity of the standard Fisher information: a counterexample
  13. A new Lyapunov functional
  14. Spherical calculus

Key Result

Theorem 1.1

Let $N \in \mathbb N$ be the number of species and $m_i>0$, $1\leq i\leq N$ their respective particle masses. Assume that the interaction is soft with $-2 >\gamma\geq -\sqrt{8}$. Further assume that the initial distributions satisfy $f^\circ_i \in L^1_{2\ell} \cap L^2_\ell \cap L \log L$ for some $ Moreover, for any $t>0$, the generalized Fisher information $\mathcal{I}\!\left(F_N\right)$ defined

Theorems & Definitions (44)

  • Theorem 1.1: Global well-posedness
  • Theorem 1.2: Non-monotonicity of the Fisher information below the critical threshold
  • Theorem 1.3: New Lyapunov functional for the infinite-mass case
  • Definition 1.4: Weighted spaces
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Two-particle dissipation identity
  • proof
  • ...and 34 more