Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case

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

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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

Bibliographical note

Funding Information:
We thank Stefan Müller, who motivated us to study this problem, for useful discussions and suggestions on the topic. The work of R.D.J. was supported by ONR (N00014-14-1-0714), AFOSR (FA9550-15-1-0207), NSF (DMREF-1629026), and the MURI Program (FA9550-18-1-0095, FA9550-16-1-0566). A.N. and J.J.L.V. acknowledge support through the CRC 1060 The mathematics of emergent effects of the University of Bonn that is funded through the German Science Foundation (DFG).

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


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


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