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<p>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.</p><p>Here we focus on the paradigmatic case of simple shear deformations and in the case of cut-off collision kernels with homogeneity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, in particular covering the case of Maxwell molecules (i.e. <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\gamma =0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and hard potentials with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(0\le \gamma &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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

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  • 发表日期：

    2025年7月15日

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

摘要

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 \(\gamma \ge 0\) γ ≥ 0 , in particular covering the case of Maxwell molecules (i.e. \(\gamma =0\) γ = 0 ) and hard potentials with \(0\le \gamma <1\) 0 ≤ 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.