Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation

Collision-Dominated Case

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

Research output: Contribution to journalArticle

Abstract

In this paper, we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, 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. We began this study in James et al. (Arch Ration Mech Anal 231(2):787–843, 2019). Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In James et al. (2019), it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper, we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case, the long-time asymptotics for the distribution of velocities is given by a time-dependent Maxwellian distribution with changing temperature.

Original languageEnglish (US)
JournalJournal of Nonlinear Science
DOIs
StatePublished - Jan 1 2019

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Long-time Asymptotics
Boltzmann equation
Boltzmann Equation
Collision
Term
Integrodifferential equations
Arches
Shear flow
Dilatation
Formal Analysis
Self-similar Solutions
Arch
Shear Flow
Ludwig Boltzmann
Integro-differential Equation
Operator
Energy
Temperature

Keywords

  • Boltzmann equation
  • Hilbert expansion
  • Homoenergetic solutions
  • Kinetic theory
  • Non-equilibrium

Cite this

Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation : Collision-Dominated Case. / James, Richard D; Nota, Alessia; Velázquez, Juan J.L.

In: Journal of Nonlinear Science, 01.01.2019.

Research output: Contribution to journalArticle

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