Vorticity layers of the 2D Navier–Stokes equations with a slip type boundary condition

Abstract

We study the asymptotic behavior, at small viscosity ε, of the Navier–Stokes equations in a 2D curved domain. The Navier–Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier–Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier–Stokes and Euler vorticity equations, we prove convergence results in the L² norm in space uniformly in time, and in the norm of H¹ in space and L² in time with rates of order ε^3/4 and ε^1/4, respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier–Stokes solution to the Euler solution in the H¹ norm uniformly in time with rate of order ε^1/4.