Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

In the present paper, we concern the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition in a two-dimensional disk to the compressible Euler equations. Due to the non-zero curvature and non-zero tangential velocity of compressible Euler solution on the boundary, new difficulties arise in the construction of Knudsen boundary layer. By employing the geometric correction, and an innovative and refined $L^2-L^\infty$ method, we establish the existence and space-decay for a truncated Knudsen boundary layer. Then, by the Hilbert expansion of multi-scales, we successfully justify the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition to the compressible Euler equations in the two-dimensional disk.