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Quantitative relaxation towards equilibrium for solutions to the Boltzmann-Fermi-Dirac equation with cutoff hard potentials

Thomas Borsoni, Bertrand Lods

Abstract

We provide the first quantitative result of convergence to equilibrium in the context of the spatially homogeneous Boltzmann-Fermi-Dirac equation associated to hard potentials interactions under angular cut-off assumption, providing an explicit - algebraic - rate of convergence to Fermi-Dirac steady solutions. This result complements the quantitative convergence result of Liu and Lu and is based upon new uniform-in-time-and-$\varepsilon$ $L^{\infty}$ bound on the solutions.

Quantitative relaxation towards equilibrium for solutions to the Boltzmann-Fermi-Dirac equation with cutoff hard potentials

Abstract

We provide the first quantitative result of convergence to equilibrium in the context of the spatially homogeneous Boltzmann-Fermi-Dirac equation associated to hard potentials interactions under angular cut-off assumption, providing an explicit - algebraic - rate of convergence to Fermi-Dirac steady solutions. This result complements the quantitative convergence result of Liu and Lu and is based upon new uniform-in-time-and- bound on the solutions.
Paper Structure (26 sections, 26 theorems, 249 equations)

This paper contains 26 sections, 26 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann-Fermi-Dirac equation
  3. Notations
  4. Relaxation to equilibrium
  5. The role of relative entropy
  6. Main results
  7. Organization of the paper
  8. Obtention of Linfty bounds
  9. A link between the Fermi-Dirac and the classical cases
  10. General estimates on Q+
  11. Estimates on Q+, Gamma and Q-
  12. Estimates on Q+
  13. Estimates on Gamma
  14. Estimates on Q-
  15. Obtention of an L2 bound
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that the collision kernel $B=B(v,v_{\ast},\sigma)$ satisfies eq:HypBB with moreover For any initial datum $f^{\rm in} \in L^1_2(\mathbb R^3)$ with $0 \leqslant f^{\rm in} \leqslant \varepsilon^{-1}$, the unique conservative (mild) solution $f^{\varepsilon}=f^{\varepsilon}(t,v)$ to (BFD Eq.) with initial datum $f^{\rm in}$ is such that where $\mathcal{M}_\varepsilon$ is the unique Fermi-D

Theorems & Definitions (57)

  • Definition 1: Fermi-Dirac statistics
  • Theorem 1: Liu & Lu (2023)
  • Remark 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Theorem 3: Uniform-in-$\varepsilon$ $L^{\infty}$ bound
  • Remark 3
  • Corollary 2
  • Theorem 4: Explicit rate of convergence to equilibrium
  • ...and 47 more