Boundary Layer Analysis of the Navier-stokes Equations with Generalized Navier Boundary Conditions

We study the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u ·n = 0, [S(u)n] tan +Au = 0, in a bounded domain in R when the viscosity, ε > 0, is small. Here, S(u) is the symmetric gradient of the velocity, u, and A is a type (1, 1) tensor on the boundary. When A = αI we obtain Navier boundary conditions, and when A is the shape operator we obtain the conditions, u · n = (curlu) × n = 0. By constructing an explicit corrector, we prove the convergence, as ε tends to zero, of the Navier-Stokes solutions to the Euler solution both in the natural energy norm and uniformly in time and space.