论文标题
Hilbert在通道中不可压缩的Euler级别中玻尔兹曼方程的扩展
Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel
论文作者
论文摘要
由于粘性和诺德森边界层的外观，对具有物理边界的玻尔兹曼方程的流体动力极限是一个具有挑战性的问题。在本文中，研究了来自玻璃体方程的流体动力极限，该方程具有镜面反射边界条件到通道中不可压缩的Euler。基于多标记的希尔伯特膨胀，分别在不同的尺度下得出了具有边界条件和兼容条件，内部解决方案，粘性和诺德森边界层的方程。然后，建立了内部解决方案，粘性和诺德森边界层的一些均匀估计。借助$ l^2-l^\ infty $框架和上面获得的均匀估计值，玻尔兹曼方程的解决方案是由带有多尺度的截短的希尔伯特膨胀构建的，因此是合理的。
The study of hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to appearance of the viscous and Knudsen boundary layers. In this paper, the hydrodynamic limit from the Boltzmann equation with specular reflection boundary condition to the incompressible Euler in a channel is investigated. Based on the multiscaled Hilbert expansion, the equations with boundary conditions and compatibility conditions for interior solutions, viscous and Knudsen boundary layers are derived under different scaling, respectively. Then some uniform estimates for the interior solutions, viscous and Knudsen boundary layers are established. With the help of $L^2-L^\infty$ 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.
