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Hypocoercivity and sub-exponential local equilibria

Emeric Bouin, Jean Dolbeault, Laurent Lafleche, Christian Schmeiser

TL;DR

The paper develops a weighted L^2 hypocoercivity framework for linear kinetic equations on the whole space without confinement, addressing sub-exponential local equilibria F(v) with 0<α<1. By combining an entropy–entropy production approach with a Nash-type macroscopic estimate and a novel weighted Poincaré inequality, it proves algebraic decay in time that matches the diffusion limit, for both Fokker–Planck (L1) and scattering (L2) collision operators. The main contributions include a new weighted coercivity result, a careful L−T splitting into dissipative and bounded parts, and a propagation theory for weighted L^2 norms that yields explicit decay rates ζ = min{d/2, k/β}. The method provides a continuous transition from sub-exponential to exponential regimes as α → 1 and does not require extra regularity assumptions, enhancing the understanding of hypocoercivity without confinement and informing diffusion-limit behavior in kinetic equations.

Abstract

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

Hypocoercivity and sub-exponential local equilibria

TL;DR

The paper develops a weighted L^2 hypocoercivity framework for linear kinetic equations on the whole space without confinement, addressing sub-exponential local equilibria F(v) with 0<α<1. By combining an entropy–entropy production approach with a Nash-type macroscopic estimate and a novel weighted Poincaré inequality, it proves algebraic decay in time that matches the diffusion limit, for both Fokker–Planck (L1) and scattering (L2) collision operators. The main contributions include a new weighted coercivity result, a careful L−T splitting into dissipative and bounded parts, and a propagation theory for weighted L^2 norms that yields explicit decay rates ζ = min{d/2, k/β}. The method provides a continuous transition from sub-exponential to exponential regimes as α → 1 and does not require extra regularity assumptions, enhancing the understanding of hypocoercivity without confinement and informing diffusion-limit behavior in kinetic equations.

Abstract

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

Paper Structure

This paper contains 13 sections, 12 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. An entropy--entropy production estimate
  3. Weighted L2 estimates
  4. A technical lemma
  5. A splitting result
  6. Proof of Proposition3
  7. Proof of Theorem 1
  8. Weighted Poincaré inequalities
  9. Continuous spectrum and weighted Poincaré inequalities
  10. A weighted Poincaré inequality with a non-classical average
  11. Algebraic decay rates for the Fokker-Planck equation
  12. Weak Poincaré inequality
  13. Weighted L2 estimates

Key Result

Theorem 1

Let $\alpha\in(0,1)$, $\beta>0$, $k>0$ and let $F$ be given by SubExponential. Assume that either $\mathsf L=\mathsf L_1$ and $\beta=2\,(1-\alpha)$, or $\mathsf L=\mathsf L_2$ and hyp:b_mass, hyp:b_bounds. Then there exists a constant ${\mathcal{C}}>0$ such that any solution $f$ of eq:main with init with rate $\zeta=\min\left\{d/2,k/\beta\right\}$ and with $\kappa>0$, which is an explicit function

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 11 more