### Abstract

In this paper we study a class of solutions of the Boltzmann equation 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. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

Original language | English (US) |
---|---|

Pages (from-to) | 787-843 |

Number of pages | 57 |

Journal | Archive For Rational Mechanics And Analysis |

Volume | 231 |

Issue number | 2 |

DOIs | |

State | Published - Feb 7 2019 |

### Fingerprint

### Cite this

*Archive For Rational Mechanics And Analysis*,

*231*(2), 787-843. https://doi.org/10.1007/s00205-018-1289-2

**Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation : Particle Velocity Distribution and Entropy.** / James, Richard D; Nota, Alessia; Velázquez, Juan J.L.

Research output: Contribution to journal › Article

*Archive For Rational Mechanics And Analysis*, vol. 231, no. 2, pp. 787-843. https://doi.org/10.1007/s00205-018-1289-2

}

TY - JOUR

T1 - Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation

T2 - Particle Velocity Distribution and Entropy

AU - James, Richard D

AU - Nota, Alessia

AU - Velázquez, Juan J.L.

PY - 2019/2/7

Y1 - 2019/2/7

N2 - In this paper we study a class of solutions of the Boltzmann equation 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. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

AB - In this paper we study a class of solutions of the Boltzmann equation 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. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

UR - http://www.scopus.com/inward/record.url?scp=85051864562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051864562&partnerID=8YFLogxK

U2 - 10.1007/s00205-018-1289-2

DO - 10.1007/s00205-018-1289-2

M3 - Article

VL - 231

SP - 787

EP - 843

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -