L 2 -compact uniform attractors for a nonautonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space

Abstract

We consider the long-time behavior of solutions for a two-dimensional nonautonomous incompressible non-Newtonian fluid with external forces in distribution space. When the external force g0(x,t) is locally uniformly integrable (see Definition 3.1) in Lloc2(R;W′), we prove the existence of L2-compact uniform attractor in space H and reveal its structure for the families of processes corresponding to the fluid. Moreover, if ∥g0∥Lb2(R;W′) is properly small, we establish the unique existence of bounded asymptotically stable solutions and give two interesting corollaries.