Complementarity, duality and symmetry in nonlinear mechanics: A FLUID PROBLEM WITH NAVIER-SLIP BOUNDARY CONDITIONS

摘要

We study the equations governing the motion of second grade fluids in a bounded domain of R{sup}d, d=2,3, with Navier slip boundary conditions with and without viscosity (averaged Euler equations). The main results concern the global existence and uniqueness of H{sup}3 solutions in dimension two and, for dimension three, local existence of H{sup}3 solutions for arbitrary initial data and global existence for small initial data and positive viscosity. The last part discusses Liapunov stability conditions for stationary solutions for the averaged Euler equations similar to the Rayleigh-Arnold stability result for the classical Euler equations. This paper presents only the main ideas of the proofs that can be found in.