A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles

Mark Wilkinson

doi:10.3934/krm.2022008

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2022, Volume 15, Issue 2: 283-315. Doi: 10.3934/krm.2022008

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A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles

Mark Wilkinson

Department of Mathematics and Physics, Nottingham Trent University, Nottingham, UK, NG1 4FQ

Received: October 2021

Early access: March 3, 2022

Published online: March 3, 2022

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Abstract

By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class $ \mathscr{C}^{1} $. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494–1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055–2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class $ \mathscr{C}^{1} $ in the classical case of hard spheres.

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Mathematics Subject Classification: Primary: 74A25; 39B22; 35Q20; 70F35.

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Figure 1. A collision configuration of two hard spheres $ \mathtt{B} $ and $ \overline{\mathtt{B}} $ in $ \mathbb{R}^{3} $, each of which is congruent to a given reference set $ \mathtt{B}_{\ast}: = \{y\in\mathbb{R}^{3}\, :\, |y|\leq \frac{1}{2}\} $. The collision parameter $ n\in\mathbb{S}^{2} $ represents the direction from the centre of mass of the unbarred sphere to that of the barred

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Figure 2. A collision configuration of two compact, convex subsets $ \mathtt{P} $ and $ \overline{\mathtt{P}} $ of $ \mathbb{R}^{3} $, each of which is congruent to a given reference set $ \mathtt{P}_{\ast} $. The matrices $ R, \overline{R}\in\mathrm{SO}(3) $ represent the orientations of the two hard particles, $ n\in\mathbb{S}^{2} $ represents the direction vector connecting the centre of mass of the unbarred particle to that of the barred, and $ d_{\beta}>0 $ denotes the distance of closest approach (8)

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Figure 3. A collision configuration of two compact, convex subsets $ \mathtt{P} $ and $ \overline{\mathtt{P}} $ of $ \mathbb{R}^{2} $, each of which is congruent to a given reference set $ \mathtt{P}_{\ast} $. The elevation angle $ \psi\in\mathbb{S}^{1} $ determines the direction vector $ e(\psi): = (\cos\psi, \sin\psi) $ directed from the centre of the unbarred particle to that of the barred, $ \vartheta, \overline{\vartheta}\in\mathbb{S}^{1} $ denote the orientations of the particles, whilst $ d_{\beta}>0 $ denotes the distance of closest approach (19)

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