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 journalArticlepeer-review

13 Scopus citations


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)
Pages (from-to)1943-1973
Number of pages31
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.


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


Dive into the research topics of 'Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case'. Together they form a unique fingerprint.

Cite this