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

Abstract

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions or on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak omega-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields) are achieved also in the strong topology of H. In particular, if a weak omega-limit set is bounded in the space V of velocity fields with square-integrable vorticity then the attraction to the set holds also in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented.