Averaging of 2D Navier-Stokes equations with singularly oscillating forces

摘要

For rho is an element of[0, 1) and epsilon > 0, the nonautonomous 2D Navier-Stokes equations with singularly oscillating external force partial derivative(t)u - upsilon Delta u + (u . del)u = -del p + g(0)(t) + epsilon(-rho) g(1) (t/epsilon), del . u = 0 are considered, together with the averaged equations partial derivative(t)u - upsilon Delta u + (u . del)u = -del p + g(0)(t) del . u = 0 formally corresponding to the limiting case epsilon = 0. Under suitable assumptions on the external force, the uniform boundedness of the related uniform global attractors A(epsilon) is established, as well as the convergence of the attractors A e of the first system to the attractor A(0) of the second one as epsilon -> 0(+). When the Grashof number of the averaged equations is small, the convergence rate of A(epsilon) to A(0) is controlled by K epsilon(1-rho)