Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

摘要

The asymptotic behavior of solutions of the three-dimensional Navier-Stokesequations is considered on bounded smooth domains with no-slip boundaryconditions or on periodic domains. Asymptotic regularity conditions arepresented to ensure that the convergence of a Leray-Hopf weak solution to itsweak omega-limit set (weak in the sense of the weak topology of the space H ofsquare-integrable divergence-free velocity fields) are achieved also in thestrong topology of H. In particular, if a weak omega-limit set is bounded inthe space V of velocity fields with square-integrable vorticity then theattraction to the set holds also in the strong topology of H. Correspondingresults for the strong convergence towards the weak global attractor of Foiasand Temam are also presented.