Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic study of the viscous layer equations and the $$L^2$$ to $$L^\infty $$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations in half-space.