Vanishing viscosity limit of the Navier-Stokes equations in a horizontally periodic strip

In this paper, we establish vanishing viscosity limit of the 2D Navier-Stokes equations in a horizontally periodic strip. On the vertical direction, the horizontal component of the velocity is subjected to two different types of boundary conditions: at the lower boundary, we give the degenerate zero boundary condition, while at the upper boundary, a small smooth perturbation of non-zero constant is prescribed. Due to this different boundary condition setting, the boundary layer effects are different near the lower and upper boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Couette flow $(Ay,0)$ for some suitable constant $A$, which is determined by using the principle of the Prandtl-Batchelor theory.