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Fractional hypocoercivity in bounded domains in the anomalous diffusion limit

Maxime Herda, Marc Pegon, Isabelle Tristani

TL;DR

The paper develops a fractional hypocoercivity framework for linear kinetic equations with heavy-tailed equilibria in bounded domains. By introducing weighted macroscopic quantities $R^\varepsilon$ and $J^\varepsilon$ and a tailored norm built from $(\mathrm{Id}-\Delta_x)^{-1}$, it obtains an $\varepsilon$-uniform coercivity estimate that drives exponential decay of solutions under general Maxwell boundary conditions and for both linear Boltzmann and Lévy--Fokker--Planck collisional models. The main contributions include a boundary-aware coercivity bound, a systematic treatment of twisted macroscopic quantities, and a rigorous route to uniform-in-$\varepsilon$ stability in the anomalous diffusion limit, with a torus variant discussed as a special case. The results extend hypocoercivity techniques to heavy-tailed equilibria and bounded domains, offering a robust tool for fractional diffusion limits in kinetic theory.

Abstract

In this paper, we provide a result of exponential stability for several dissipative linear kinetic equations with heavy-tailed equilibria. The approach, inspired by the so-called $L^2$-hypocoercivity method, is robust enough to provide estimates that are uniform in the anomalous diffusion limit. Moreover, it is able to deal with bounded domains with periodic boundary condition or general Maxwell boundary condition (from the pure specular to the pure diffusive case). In addition, our framework accommodates linear collisional operators that act simultaneously on the velocity and spatial variables.

Fractional hypocoercivity in bounded domains in the anomalous diffusion limit

TL;DR

The paper develops a fractional hypocoercivity framework for linear kinetic equations with heavy-tailed equilibria in bounded domains. By introducing weighted macroscopic quantities and and a tailored norm built from , it obtains an -uniform coercivity estimate that drives exponential decay of solutions under general Maxwell boundary conditions and for both linear Boltzmann and Lévy--Fokker--Planck collisional models. The main contributions include a boundary-aware coercivity bound, a systematic treatment of twisted macroscopic quantities, and a rigorous route to uniform-in- stability in the anomalous diffusion limit, with a torus variant discussed as a special case. The results extend hypocoercivity techniques to heavy-tailed equilibria and bounded domains, offering a robust tool for fractional diffusion limits in kinetic theory.

Abstract

In this paper, we provide a result of exponential stability for several dissipative linear kinetic equations with heavy-tailed equilibria. The approach, inspired by the so-called -hypocoercivity method, is robust enough to provide estimates that are uniform in the anomalous diffusion limit. Moreover, it is able to deal with bounded domains with periodic boundary condition or general Maxwell boundary condition (from the pure specular to the pure diffusive case). In addition, our framework accommodates linear collisional operators that act simultaneously on the velocity and spatial variables.
Paper Structure (13 sections, 8 theorems, 99 equations)

This paper contains 13 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Functional framework
  4. Main result
  5. Preliminaries
  6. Elliptic problems
  7. Coercivity estimate
  8. Boundary terms
  9. New macroscopic quantities
  10. Fractional hypocoercivity
  11. Definition of the new norm
  12. Proof of Theorem \ref{['theo:main']}
  13. The case of the torus

Key Result

Theorem 1.2

Consider $\varepsilon \in (0,1]$. There exists a scalar product $\langle\!\langle \cdot , \cdot \rangle \! \rangle_{\varepsilon}$ on the space $\mathcal{H}$ so that the associated norm $| | | \cdot | | |_\varepsilon$ is equivalent to the usual norm $\| \cdot \|_{\mathcal{H} }$ uniformly in $\varepsi for any $f \in \mathrm{Dom}(\Lambda^\varepsilon)$ satisfying the boundary condition in eq:dtf=Lf an

Theorems & Definitions (15)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 5 more