The effective boundary conditions for vector fields on domains with rough boundaries: Applications to fluid mechanics

Consider a vicous incompressible fluid occupying a bounded domain Ω ⊂ R. In the Eulerean reference system, the motion of the fluid is completely determined by the velocity field u = u(t, x) a vector valued function of the time t and the spatial position x ∈ Ω. Under the hypothesis of impermeability of the boundary, the velocity satisfies u · n|∂Ω = 0, (1.1) where the symbol n stands for the outer normal vector. In addition to (1.1), the widely accepted hypothesis asserts there is no relative motion between a viscous fluid and the rigid wall represented by ∂Ω, meaning