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<p>We consider the existence of steady rarefied flows of polyatomic gas between two parallel condensed phases, where evaporation and condensation processes occur. To this end, we study the existence problem of stationary solutions in a one-dimensional slab for the polyatomic Boltzmann equation, which takes into account the effect of internal energy in the collision process of the gas molecules. We show that, under suitable norm bound assumptions on the boundary condition functions, there exists a unique mild solution to the stationary polyatomic Boltzmann equation when the slab is sufficiently small. This is based on various norm estimates - singular estimates, hyperplane estimates - of the collision operator, for which genuinely polyatomic techniques must be employed. The key observation is that there is a polyatomic regularizing effect on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Q^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>, which leads to a refined estimate of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Q^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> that does not appear in the monatomic case. This regularizing effect allows us to establish existence results under less restrictive conditions on the boundary data than in the monatomic case.</p>

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Stationary Boltzmann Equation for Polyatomic Gases in a slab

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Ki Nam Hong,
  • Marwa Shahine,
  • Seok-Bae Yun

摘要

We consider the existence of steady rarefied flows of polyatomic gas between two parallel condensed phases, where evaporation and condensation processes occur. To this end, we study the existence problem of stationary solutions in a one-dimensional slab for the polyatomic Boltzmann equation, which takes into account the effect of internal energy in the collision process of the gas molecules. We show that, under suitable norm bound assumptions on the boundary condition functions, there exists a unique mild solution to the stationary polyatomic Boltzmann equation when the slab is sufficiently small. This is based on various norm estimates - singular estimates, hyperplane estimates - of the collision operator, for which genuinely polyatomic techniques must be employed. The key observation is that there is a polyatomic regularizing effect on \(Q^+\) Q + , which leads to a refined estimate of \(Q^+\) Q + that does not appear in the monatomic case. This regularizing effect allows us to establish existence results under less restrictive conditions on the boundary data than in the monatomic case.