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 | |
State | Published - Oct 1 2019 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Keywords
- Boltzmann equation
- Hilbert expansion
- Homoenergetic solutions
- Kinetic theory
- Non-equilibrium