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Passing to the limit in fuzzy Boltzmann equations

Matthias Erbar, Zihui He

TL;DR

The paper studies a fuzzy version of the inhomogeneous Boltzmann equation, where collisions are spatially delocalised via a kernel that is modulated by a scale parameter. It proves that as the spatial kernel concentrates to a Dirac mass, the fuzzy solutions converge (up to subsequences) to renormalised solutions of the classical inhomogeneous Boltzmann equation, while preserving entropy dissipation. The approach combines existence and compactness theory for the fuzzy model with velocity-averaging techniques and an abstract strong-compactness framework (à la DiPerna–Lions and Lions–Lions), ultimately yielding strong convergence in $C([0,T];L^1)$ and identification of the limit as a renormalised solution. The results provide a rigorous link between a GENERIC-inspired fuzzy model and the classical kinetic theory, offering a robust pathway to handle delocalised interactions and complex collision structures in inhomogeneous settings.

Abstract

We study a fuzzy Boltzmann equation, where collisions are delocalised and modulated by a spatial kernel. We show that as the spatial kernel converges to a delta distribution, the solutions to these equations converge to renormalised solutions of the inhomogeneous Boltzmann equations.

Passing to the limit in fuzzy Boltzmann equations

TL;DR

The paper studies a fuzzy version of the inhomogeneous Boltzmann equation, where collisions are spatially delocalised via a kernel that is modulated by a scale parameter. It proves that as the spatial kernel concentrates to a Dirac mass, the fuzzy solutions converge (up to subsequences) to renormalised solutions of the classical inhomogeneous Boltzmann equation, while preserving entropy dissipation. The approach combines existence and compactness theory for the fuzzy model with velocity-averaging techniques and an abstract strong-compactness framework (à la DiPerna–Lions and Lions–Lions), ultimately yielding strong convergence in and identification of the limit as a renormalised solution. The results provide a rigorous link between a GENERIC-inspired fuzzy model and the classical kinetic theory, offering a robust pathway to handle delocalised interactions and complex collision structures in inhomogeneous settings.

Abstract

We study a fuzzy Boltzmann equation, where collisions are delocalised and modulated by a spatial kernel. We show that as the spatial kernel converges to a delta distribution, the solutions to these equations converge to renormalised solutions of the inhomogeneous Boltzmann equations.
Paper Structure (13 sections, 20 theorems, 158 equations)

This paper contains 13 sections, 20 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Fuzzy Boltzmann equation
  3. Main results
  4. Solvability and compactness results
  5. Definition and properties of solutions
  6. Weak compactness results
  7. Applications of velocity averaging
  8. Strong convergence results
  9. Convergence in measure
  10. Proof of \ref{['conv:f-sigma-M:M']}
  11. Proof of \ref{['conv:f-sigma-M:sigma']}
  12. Strong convergence
  13. Solvability results

Key Result

Theorem 1.1

Let $f_0\in L^1_{2,2}(\mathbb{R}^{2d})$ such that $\mathcal{H}(f_0)<+\infty$. Let the collision kernel $B$ satisfy the Assumption CK. Let $f^\sigma\in C([0,T];L^1(\mathbb{R}^{2d})$ be a weak solution of the fuzzy Boltzmann equation pre:FBE such that for some $C=C(T)>0$. Then there exists a renormalised solution $f\in C([0,T];L^1(\mathbb{R}^{2d}))\cap L^\infty([0,T]; L^1_{2,2}(\mathbb{R}^{2d}))$ o

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8: Dunford--Pettis
  • Lemma 2.9
  • ...and 26 more