Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

Feimin Huang; Jing Ouyang; Yong Wang

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arXiv · 2504.14932

Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

Feimin Huang·Jing Ouyang·Yong Wang

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Abstract

In the present paper, we concern the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition in a two-dimensional disk to the compressible Euler equations. Due to the non-zero curvature and non-zero tangential velocity of compressible Euler solution on the boundary, new difficulties arise in the construction of Knudsen boundary layer. By employing the geometric correction, and an innovative and refined $L^2-L^\infty$ method, we establish the existence and space-decay for a truncated Knudsen boundary layer. Then, by the Hilbert expansion of multi-scales, we successfully justify the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition to the compressible Euler equations in the two-dimensional disk.

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Feimin Huang, Jing Ouyang, Yong Wang. 2025-04-21. Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition. https://arxiv.org/abs/2504.14932

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