Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays

Abstract

In this paper we strengthen some results on the existence and properties of pullback attractors for a 2D Navier-Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271-297]. Actually, we prove that under suitable assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Moreover, thanks to regularity results, the attraction from different phase spaces also happens in C([−h, 0]; V ). Finally, from comparison results of attractors, and under an additional hypothesis, we establish that all these families of attractors are in fact the same object.

Stability results for 2D Navier–Stokes equations with unbounded delay

“…Now we provide some examples of (unbounded) delay forcing terms which can be set within our general set-up (see [22,19,20,18,21]).…”

“…The analysis of the Navier-Stokes equations with hereditary terms was initiated by Caraballo and Real in [14], and developed in [11,12,13,7,15,8,9,10,5,6], where several issues have been investigated: the existence and uniqueness of solution, stationary solution, the existence of attractors (global, pullback and random ones) and the local exponential stability of state-steady solution of Navier-Stokes models with several types of delay (constant, bounded variable delay as well as bounded distributed delay). In the papers [22,33,29,34,30,31,35,19,32,20,18,21] the authors have discussed the asymptotic behavior and regularity of solutions of 2D Navier-Stokes equations (and 3D-variations of Navier-Stokes models) with delay (finite and infinite). Wei and Zhang [39] have obtained the exponential stability and almost sure exponential stability of the weak solution for stochastic 2D Navier-Stokes equations with bounded variable delays by using the approach proposed in [14,6].…”

Some regularity results for a double time-delayed 2D-Navier-Stokes model

“…Last claim is simpler. If u τ ∈ V, it only requires integration in (8) in [τ, t] and application of the Gronwall lemma. The details are omitted for brevity.…”

The pullback‐ attractors for the 3D Kelvin‐Voigt‐Brinkman‐Forchheimer system with delay

“…In addition, the delay terms are natural to appear in fluid flow, and many results had been derived since the Navier-Stokes system with delay was constructed in Krasovskii, 15 involving the long-time behavior of solutions and the existence of attractors, we can refer to previous studies. [16][17][18][19][20][21][22][23] We first give some useful results relating to function spaces and inequalities in part 2. In part 3, the well-posedness of solutions to the system (1.1) is derived, and in part 4, we establish the pullback- asymptotical compactness of the solution process.…”