Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation: Particle Velocity Distribution and Entropy

Richard D James, Alessia Nota, Juan J.L. Velázquez

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In this paper we study a class of solutions of the Boltzmann equation which have the form f (x, v, t) = g (v − L (t) x, t) where L (t) = A (I + tA)−1 with the matrix A describing a shear flow or a dilatation or a combination of both. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

Original languageEnglish (US)
Pages (from-to)787-843
Number of pages57
JournalArchive For Rational Mechanics And Analysis
Volume231
Issue number2
DOIs
StatePublished - Feb 7 2019

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Boltzmann equation
Velocity Distribution
Velocity distribution
Boltzmann Equation
Entropy
Molecules
Long-time Asymptotics
H-function
Dilatation
Self-similar Solutions
Asymptotics of Solutions
Shear flow
Shear Flow
Homogeneity
Existence of Solutions
Collision
kernel
Profile
Class
Temperature

Cite this

Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation : Particle Velocity Distribution and Entropy. / James, Richard D; Nota, Alessia; Velázquez, Juan J.L.

In: Archive For Rational Mechanics And Analysis, Vol. 231, No. 2, 07.02.2019, p. 787-843.

Research output: Contribution to journalArticle

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