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

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

doi:10.1007/s00332-019-09535-6

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

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

Aerospace Engineering and Mechanics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

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 language	English (US)
Pages (from-to)	1943-1973
Number of pages	31
Journal	Journal of Nonlinear Science
Volume	29
Issue number	5
DOIs	https://doi.org/10.1007/s00332-019-09535-6
State	Published - 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.

Keywords

Boltzmann equation
Hilbert expansion
Homoenergetic solutions
Kinetic theory
Non-equilibrium

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10.1007/s00332-019-09535-6

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title = "Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case",

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.",

keywords = "Boltzmann equation, Hilbert expansion, Homoenergetic solutions, Kinetic theory, Non-equilibrium",

author = "James, {Richard D.} and Alessia Nota and Vel{\'a}zquez, {Juan J.L.}",

note = "Funding Information: We thank Stefan M{\"u}ller, 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: {\textcopyright} 2019, Springer Science+Business Media, LLC, part of Springer Nature.",

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N1 - 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.

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N2 - 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.

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

Abstract

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