The Incompressible Navier-Stokes System with Time-Dependent Robin-Type Boundary Conditions

We show that the incompressible 3D Navier-Stokes system in a l(1,1) bounded domain or a bounded convex domain Omega with a non penetration condition nu.u = 0 at the boundary partial derivative Omega together with a time-dependent Robin boundary condition of the type nu x curl u = beta(t)u on partial derivative Omega admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.