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<p>Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in Kim and Lee (Commun Pure Appl Math 77(4):2331–2386, 2024), where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^{\frac{1}{2}-}_{x,v}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>v</mi> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> Hölder regularity was proven. In this paper, we introduce a novel <i>dynamical singular regime integration</i> methodology to establish the optimal <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^{\frac{1}{2}}_{x,v}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>v</mi> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msubsup> </math></EquationSource> </InlineEquation> regularity for the Boltzmann equation past a convex obstacle.</p>

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Optimal \(C^{\frac{1}{2}}\) Regularity of the Boltzmann Equation in Non-Convex Domains

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  • 发表日期：

    2026年5月6日

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  • Gayoung An,
  • Donghyun Lee

摘要

Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in Kim and Lee (Commun Pure Appl Math 77(4):2331–2386, 2024), where \(C^{\frac{1}{2}-}_{x,v}\) C x , v 1 2 - Hölder regularity was proven. In this paper, we introduce a novel dynamical singular regime integration methodology to establish the optimal \(C^{\frac{1}{2}}_{x,v}\) C x , v 1 2 regularity for the Boltzmann equation past a convex obstacle.