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Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded domain

Cyril Imbert, Amélie Loher

TL;DR

The paper addresses the challenge of establishing polynomial decay in the velocity variable for the space-inhomogeneous non-cutoff Boltzmann equation in a bounded domain under hydrodynamic bounds. It develops a framework of Truncated Convex Inequalities and a barrier technique to control the positive part of (f−A) and to generate decay, handling both hard and moderately soft potentials. The authors prove that, under suitable mass, energy, and entropy constraints, suitable weak subsolutions exhibit polynomial decay of the form $f(t,x,v)\le C_q(1+t)^{-\beta_q}\langle v\rangle^{-q}$ for all $q\ge0$, with explicit dependence of $\beta_q$ on dimension and collision parameters, and they establish generation of velocity moments even in the soft-potential regime. The work extends previous homogeneous results to bounded domains, contributes novel coercivity and error-term analyses (including the important “good extra term”), and provides a pathway toward conditional regularity in bounded settings, with implications for long-time behavior and Maxwellian bounds under the given hypotheses.

Abstract

This work is concerned with the appearance of decay bounds in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff posed in a domain in the case of hard and moderately soft potentials. Such bounds are derived for general non-negative suitable weak subsolutions. These estimates hold true as long as mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The proof relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced by F. Golse, L. Silvestre and the first author (2023). To the best of our knowledge, the generation of arbitrary polynomial decay in the velocity variable for the Boltzmann equation without cutoff is new in the case of soft potentials, even for classical solutions.

Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded domain

TL;DR

The paper addresses the challenge of establishing polynomial decay in the velocity variable for the space-inhomogeneous non-cutoff Boltzmann equation in a bounded domain under hydrodynamic bounds. It develops a framework of Truncated Convex Inequalities and a barrier technique to control the positive part of (f−A) and to generate decay, handling both hard and moderately soft potentials. The authors prove that, under suitable mass, energy, and entropy constraints, suitable weak subsolutions exhibit polynomial decay of the form for all , with explicit dependence of on dimension and collision parameters, and they establish generation of velocity moments even in the soft-potential regime. The work extends previous homogeneous results to bounded domains, contributes novel coercivity and error-term analyses (including the important “good extra term”), and provides a pathway toward conditional regularity in bounded settings, with implications for long-time behavior and Maxwellian bounds under the given hypotheses.

Abstract

This work is concerned with the appearance of decay bounds in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff posed in a domain in the case of hard and moderately soft potentials. Such bounds are derived for general non-negative suitable weak subsolutions. These estimates hold true as long as mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The proof relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced by F. Golse, L. Silvestre and the first author (2023). To the best of our knowledge, the generation of arbitrary polynomial decay in the velocity variable for the Boltzmann equation without cutoff is new in the case of soft potentials, even for classical solutions.
Paper Structure (71 sections, 42 theorems, 356 equations)

This paper contains 71 sections, 42 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. Hydrodynamical quantities
  4. The collision operator
  5. Truncated Convex Inequalities
  6. Suitable weak subsolutions
  7. Main result
  8. Comments
  9. Conditional decay estimates.
  10. Barriers and energy methods.
  11. Truncated Convex Inequalities.
  12. Coercive terms.
  13. Error terms.
  14. Generation of pointwise decay.
  15. Review of literature
  16. ...and 56 more sections

Key Result

Theorem 1.6

Let the parameters $\gamma \in (-d, 1]$ and $s \in (0, 1)$ of the non-cutoff collision kernel satisfy $\gamma + 2s \in (0,1)$, Let $f_{\mathrm{in}}$ (and $f_b$) denote the initial data (and the boundary data in the in-flow case). Let $T \in (0,+\infty]$ and $f$ be a suitable weak subsolution of the where $D(f) \ge 0$ denotes the entropy production. We also assume that $f$ satisfies Condition e:hy

Theorems & Definitions (94)

  • Definition 1.1: Suitable weak subsolutions
  • Remark 1.2: General convex functions
  • Remark 1.3: Link with renormalized solutions
  • Remark 1.4: Positive terms
  • Remark 1.5: Error terms
  • Theorem 1.6: Conditional decay estimates
  • Remark 1.7: Entropy production
  • Remark 1.8: Integrability of solutions
  • Remark 1.9: Algebraic time decay in $v$
  • Remark 1.10: Large times
  • ...and 84 more