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Semi-local behaviour of non-local hypoelliptic equations: Boltzmann

Amélie Loher

TL;DR

The paper proves a semi-local Strong Harnack inequality for the Boltzmann equation with moderately soft, non-cutoff potentials under hydrodynamic bounds, revealing semi-local control by exploiting the average divergence structure of the collision operator. It develops a conditional framework in which the nonlocal operator has a symmetric core and a low-order antisymmetric part, enabling a combination of tail bounds, L^1-to-L^∞ smoothing, and fractional Kolmogorov regularity to yield the SHarnack inequality. The results lead to bounds on the fundamental solution of the linearised Boltzmann equation, including polynomial upper bounds and exponential lower bounds, via Aronson-type arguments adapted to nonlocal hypoelliptic operators in divergence form on average. The work fills gaps in previous proofs, extends weak Harnack and Schauder-type regularity to a strong, tail-aware estimate, and enhances understanding of nonlocal kinetic equations beyond cutoff models. Overall, the findings provide a rigorous pathway from nonlinear Boltzmann dynamics to quantitative fundamental solution bounds under physically relevant, globally defined velocity behavior.

Abstract

The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, Séminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in [arXiv:2404.05612] does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation.

Semi-local behaviour of non-local hypoelliptic equations: Boltzmann

TL;DR

The paper proves a semi-local Strong Harnack inequality for the Boltzmann equation with moderately soft, non-cutoff potentials under hydrodynamic bounds, revealing semi-local control by exploiting the average divergence structure of the collision operator. It develops a conditional framework in which the nonlocal operator has a symmetric core and a low-order antisymmetric part, enabling a combination of tail bounds, L^1-to-L^∞ smoothing, and fractional Kolmogorov regularity to yield the SHarnack inequality. The results lead to bounds on the fundamental solution of the linearised Boltzmann equation, including polynomial upper bounds and exponential lower bounds, via Aronson-type arguments adapted to nonlocal hypoelliptic operators in divergence form on average. The work fills gaps in previous proofs, extends weak Harnack and Schauder-type regularity to a strong, tail-aware estimate, and enhances understanding of nonlocal kinetic equations beyond cutoff models. Overall, the findings provide a rigorous pathway from nonlinear Boltzmann dynamics to quantitative fundamental solution bounds under physically relevant, globally defined velocity behavior.

Abstract

The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, Séminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in [arXiv:2404.05612] does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation.
Paper Structure (21 sections, 9 theorems, 118 equations)

This paper contains 21 sections, 9 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Boltzmann collision operator
  3. Integro-differential equation with non-negative source
  4. Integro-differential equation in divergence form on average
  5. Conditional regime
  6. Main result
  7. Setup
  8. Notion of solutions
  9. Ellipticity class
  10. On the non-divergent part of the kernel
  11. Notation
  12. Tail bound on upper level sets
  13. Linear $L^1$ to $L^\infty$ bound
  14. Fractional Kolmogorov
  15. Proof of Proposition \ref{['prop:L1-Linfty']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $T > 0$. Suppose $0 \leq f$ satisfies eq:hydro. Let $f$ solve eq:boltzmann in $(0,T) \times {\mathbb R}^d \times {\mathbb R}^d$ in the sense of Definition def:sol with a cross section $B$ given by eq:B with $0 \leq \gamma +2s \leq 2$, and with initial data $f_0(x, v) = f(0, x, v)$, which in case where $0 < \tau_0 < \tau_1 < \tau_2 < \tau_3$ such that $\tau_1 - \tau_0 = \tau_2 - \tau_3 = (\frac

Theorems & Definitions (15)

  • Theorem 1.1: Strong Harnack for Boltzmann
  • Definition 2.1: Solutions
  • Proposition 3.1: Non-local-to-local bound
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:prelim']}
  • proof : Proof of Proposition \ref{['prop:tail-bound']}
  • Proposition 4.1: $L^1$-$L^\infty$ bound
  • Proposition 4.2
  • proof : Proof of Theorem \ref{['thm:sh-be']}
  • Theorem 5.1: Polynomial upper bounds on the fundamental solution
  • ...and 5 more