Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations

Both existence and backwards topological property of pullback attractors are discussed for 3D Navier-Stokes equations with a nonlinear damping and a non-autonomous force. A pullback attractor is obtained in a square integrable space if the order of damping is larger than three and further in a Sobolev space if the order belongs to (3, 5), the latter of which improves the best range [7/2, 5) given in literatures so far. The new hypotheses on the force used here are weaker than those given in literatures. More importantly, the obtained attractor is shown to be backwards compact, i.e. the union of attractors over the past time is pre-compact. This result is a successful application of some new abstract criteria on backwards compact attractors if an evolution process is backwards pullback limit-set compact or equivalently backwards pullback flattening.