Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions

Abstract

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 R3 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.