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<p>In this paper, we focus on the Cauchy problem for the Boltzmann equation with a deformation matrix A describing a shear flow. For different choices of the matrix A and various homogeneities of the collision kernel, it is observed that the long-time behavior of the velocity distribution for the corresponding self-similar solutions varies. The main results are twofold: (1) For a general deformation force with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(trA&lt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mi>r</mi> <mi>A</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the well-posedness and large-time behavior for all hard potentials; (2) In the case of planar shear flow with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(trA\ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mi>r</mi> <mi>A</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the stability of the stationary solution for the Maxwell molecules is proved.</p>

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The stability of the Boltzmann equation with deformations

  • 期刊论文

  • 发表日期：

    2025年9月24日

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  • Shuangqian Liu,
  • Yating Wang,
  • Anita Yang,
  • Xueying Zhang

摘要

In this paper, we focus on the Cauchy problem for the Boltzmann equation with a deformation matrix A describing a shear flow. For different choices of the matrix A and various homogeneities of the collision kernel, it is observed that the long-time behavior of the velocity distribution for the corresponding self-similar solutions varies. The main results are twofold: (1) For a general deformation force with \(trA<0\) t r A < 0 , we establish the well-posedness and large-time behavior for all hard potentials; (2) In the case of planar shear flow with \(trA\ge 0\) t r A ≥ 0 , the stability of the stationary solution for the Maxwell molecules is proved.