Uniform attractors for three-dimensional Navier–Stokes equations with nonlinear damping

Abstract

This paper is concerned with the three-dimensional non-autonomous Navier–Stokes equation with nonlinear damping in 3D bounded domains. When the external force f0(x,t) is translation compact in Lloc2(R;H), α>0, 72≤β≤5 and initial data uτ∈V, we give a series of uniform estimates on the solutions. Based on these estimates, we prove the family of processes {Uf(t,τ)}, f∈H(f0), is (V×H(f0),V)-continuous. At the same time, by making use of Ascoli–Arzela theorem, we find {Uf(t,τ)}, f∈H(f0), is (V,H2(Ω))-uniformly compact. So, using semiprocess theory, we obtain the existence of (V,V)-uniform attractor and (V,H2(Ω))-uniform attractor. And we prove the (V,V)-uniform attractor is actually the (V,H2(Ω))-uniform attractor.