In a bounded domain with smooth boundary (which can be considered as a smooth sub-manifold of R^3), we consider the Boltzmann equation with general Maxwell boundary condition---linear combination of specular reflection and diffusive absorption. We analyze the kinetic (Knudsen layer) and fluid (viscous layer) coupled boundary layers in both acoustic and incompressible regimes, in which the boundary layers behave significantly different. The existence and damping properties of these kinetic-fluid layers depends on the relative size of accommodation number and Kundsen number, and the differential geometric property of the boundary (the second fundamental form.) As applications, first we justify the incompressible Navier-Stokes-Fourier limit of the Boltzmann equation with Dirichlet, Navier, and diffusive boundary conditions respectively, depending on the relative size of accommodation number and Kundsen number. Using the damping property of the boundary layer in acoustic regime, we proved the convergence is strong. The second application is that we derive and justified the higher order acoustic approximation of the Boltzmann equation.

相关理论研究的最终目标是：从统计力学(微观或介观)出发推导流体力学方程组(即 Naveir-Stokes方程)。也即给Naveir-Stokes方程一个严格的统计理学的推导。这是伟大而艰难的计划。陈天权教授(英文专著)从统计力学中的Liuville方程出发，即不用分子混沌假设，定性地得到结论：Naveir-Stokes方程中的粘性部分应予修正。但具体修正形式需要艰难庞大的计算。法国高师Golse 等人从Boltzmann方程(即相当于承认分子混沌假设）出发证明了Boltzmann方程当粒子平均自由程趋于零时趋于Naveir-Stokes方程。 Dave Levermore的框架似乎介于二者之间，吸收了各自的部分优点。江宁的(合作)工作即于此有关。他与 Masmoudi教授在kinetic-fluid boundary layers方面的一个的工作彻底解决了这个领域的一个open问题。