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Homoenergetic solutions for the Rayleigh-Boltzmann equation: existence of a stationary non-equilibrium solution

Nicola Miele, Alessia Nota, Juan J. L. Velázquez

TL;DR

The paper analyzes homoenergetic solutions of the linear Boltzmann-Rayleigh equation under simple shear, revealing how the long-time behavior depends on the collision kernel homogeneity $\gamma$. It proves well-posedness in the space of non-negative Radon measures for $0\le \gamma<1$, using adjoint problems and Markov semigroups, and then shows the existence of a stationary non-equilibrium state driven by the shear. For Maxwell molecules ($\gamma=0$) a stationary state exists when the shear is sufficiently small, while for larger shear the moment matrix can grow exponentially, precluding stationarity. For hard potentials with $0<\gamma<1$, a stationary non-equilibrium solution exists without a smallness constraint on the shear parameter, thanks to the global boundedness of the second moment. These results highlight a balance between shear-induced transport and collisional relaxation in a linear/open-system setting, contrasting with the nonlinear Boltzmann case where stationary self-similar states behave differently.

Abstract

In this paper we consider a particular class of solutions of the linear Boltzmann-Rayleigh equation, known in the nonlinear setting as Homoenergetic solutions. These solutions describe the dynamics of Boltzmann gases under the effect of different mechanical deformations. Therefore, the long-time behaviour of these solutions cannot be described by Maxwellian distributions and it strongly depends on the homogeneity of the collision kernel of the equation. Here we focus on the paradigmatic case of simple shear deformations and in the case of cut-off collision kernels with homogeneity $γ\geq 0$, in particular covering the case of Maxwell molecules (i.e. $γ=0$) and hard potentials with $0\leq γ<1$. We first prove a well-posedness result for this class of solutions in the space of non-negative Radon measures and then we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the shear deformation. In the case of Maxwell molecules we prove that there is a different behaviour of the solutions for small and large values of the shear parameter.

Homoenergetic solutions for the Rayleigh-Boltzmann equation: existence of a stationary non-equilibrium solution

TL;DR

The paper analyzes homoenergetic solutions of the linear Boltzmann-Rayleigh equation under simple shear, revealing how the long-time behavior depends on the collision kernel homogeneity . It proves well-posedness in the space of non-negative Radon measures for , using adjoint problems and Markov semigroups, and then shows the existence of a stationary non-equilibrium state driven by the shear. For Maxwell molecules () a stationary state exists when the shear is sufficiently small, while for larger shear the moment matrix can grow exponentially, precluding stationarity. For hard potentials with , a stationary non-equilibrium solution exists without a smallness constraint on the shear parameter, thanks to the global boundedness of the second moment. These results highlight a balance between shear-induced transport and collisional relaxation in a linear/open-system setting, contrasting with the nonlinear Boltzmann case where stationary self-similar states behave differently.

Abstract

In this paper we consider a particular class of solutions of the linear Boltzmann-Rayleigh equation, known in the nonlinear setting as Homoenergetic solutions. These solutions describe the dynamics of Boltzmann gases under the effect of different mechanical deformations. Therefore, the long-time behaviour of these solutions cannot be described by Maxwellian distributions and it strongly depends on the homogeneity of the collision kernel of the equation. Here we focus on the paradigmatic case of simple shear deformations and in the case of cut-off collision kernels with homogeneity , in particular covering the case of Maxwell molecules (i.e. ) and hard potentials with . We first prove a well-posedness result for this class of solutions in the space of non-negative Radon measures and then we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the shear deformation. In the case of Maxwell molecules we prove that there is a different behaviour of the solutions for small and large values of the shear parameter.
Paper Structure (14 sections, 23 theorems, 219 equations)

This paper contains 14 sections, 23 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Well-Posedness theory for the Boltzmann-Rayleigh equation with simple shear deformations.
  3. Well-Posedness theory in the case of pseudo Maxwellian molecules (i.e. $\gamma=0$)
  4. A comment on the well-posedness result for \ref{['eq:Cauchy2']} by means of Fourier transform
  5. Well-posedness theory in the case of collision kernels with homogeneity $\gamma>0$
  6. Existence of a Stationary non-equilibrium solution for the Boltzmann-Rayleigh equation with simple shear deformations.
  7. The case of pseudo Maxwell molecules
  8. Moments Equation for Maxwell Molecules
  9. Existence of a stationary solution (Proof of Theorem \ref{['thm:ssMM']})
  10. Non existence of a stationary solution for large values of the shear parameter $K$
  11. The case of collision kernels with homogeneity $\gamma> 0$
  12. Existence of a stationary solution (Proof of Theorem \ref{['thm:stationary>0']})
  13. A remark on the consistency of homoenergetic solution in the linear setting
  14. Moment equations (cf. Proposition \ref{['lem:derM']}) in the case of constant collision kernel

Key Result

Theorem 2.1

Suppose that $f_0 \in \mathscr{M}_{+}\left(\mathbb{R}^3 \right)$ and that $B$ satisfies eq:AssB1. Then, there exists a unique weak solution $f \in C\left( [0,+\infty),\mathscr{M}_{+}\left(\mathbb{R}^3 \right) \right)$ of eq:Cauchy2 in the sense of Definition def:weakSol.

Theorems & Definitions (55)

  • Definition 1
  • Remark 1
  • Theorem 2.1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Definition 4
  • Theorem 2.2: Hille-Yosida
  • Proposition 2.3
  • ...and 45 more