### 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) |
---|---|

Journal | Journal of Nonlinear Science |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Nonlinear Science*. https://doi.org/10.1007/s00332-019-09535-6

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation

T2 - Collision-Dominated Case

AU - James, Richard D

AU - Nota, Alessia

AU - Velázquez, Juan J.L.

PY - 2019/1/1

Y1 - 2019/1/1

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.

AB - 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.

KW - Boltzmann equation

KW - Hilbert expansion

KW - Homoenergetic solutions

KW - Kinetic theory

KW - Non-equilibrium

UR - http://www.scopus.com/inward/record.url?scp=85062702845&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062702845&partnerID=8YFLogxK

U2 - 10.1007/s00332-019-09535-6

DO - 10.1007/s00332-019-09535-6

M3 - Article

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

ER -