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The Hilbert expansion of the Boltzmann equation in the incompressible euler level in a channel

Science China Mathematics 2025年第1期68卷 39-88页

作者： Feimin Huang Weiqiang Wang Yong Wang Feng Xiao Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education) School of Mathematics and StatisticsHunan Normal University

The study of the hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to the appearance of the viscous and Knudsen boundary *** this paper,the hydrodynamic limit from the Boltzmann equation with the specular reflection boundary condition to the incompressible euler equations in a channel is *** on the multi-scaled Hilbert expansion,the equations with boundary conditions and compatibility conditions for interior solutions,and viscous and Knudsen boundary layers are derived under different scaling,***,some uniform estimates for the interior solutions,and viscous and Knudsen boundary layers are *** the help of the L2-L∞framework and the uniform estimates obtained above,the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales,and hence the hydrodynamic limit in the incompressible euler level is justified.

关键词： Boltzmann equation incompressible euler equations hydrodynamic limit Hilbert expansion specular reflection boundary condition viscous boundary layer Knudsen boundary layer

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On the local wellposedness of 3-D water wave problem with vorticity

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Science China Mathematics 2007年第8期50卷 1065-1077页

作者： Ping ZHANG~1 Zhi-fei ZHANG~2 1 Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China 2 School of Mathematical Sciences,Peking University,Beijing 100871,China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China School of Mathematical Sciences Peking University Beijing China

In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that $\frac{{\partial p}}{{\partial n}}(\xi )\left| {_{t = 0} } \right. \leqslant - 2c_0 < 0$ being restricted to the initial surface.

关键词： water-waves free boundary incompressible euler equations primary 35Q35 76B03 secondary 35J67 35L80

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