Boundary Layers and Hydrodynamic Limits of Boltzmann Equation (i): Incompressible Navier-stokes-fourier Limit

We establish an incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation (with a general collision kernel) considered over a bounded domain. Appropriately scaled families of DiPerna-Lions renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number ε goes to zero. Every limit point is a weak solution to a NavierStokes-Fourier system with different types of fluid boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately which means that they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. These acoustic waves are known to oscillate faster and faster when the Knudsen number ε goes to zero and to prevent strong convergence in the periodic or the whole space case. This extends the work of Golse and Saint-Raymond [18, 19] and Levermore and Masmoudi [27] to the case of a bounded domain. The case of a bounded domain was already considered by Masmoudi and Saint-Raymond [33] for the (linear) Stokes-Fourier limit and without studying the damping of the acoustic waves.