Dynamics of 2D Incompressible Non-autonomous Navier–Stokes Equations on Lipschitz-like Domains

Abstract

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation with non-homogeneous boundary condition on Lipschitz-like domain. With the presence of a time-dependent external force f (t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimate on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is the sum of a stationary force and a non-autonomous perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we also investigate the regularity of these attractors when smoother initial data is given. Our results are new even in the case of smooth domains.

Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains

“…The tempered pullback attractors here contain the ω-limit sets and all equilibriums for the background flow, which differs from the corresponding results in bounded smooth domain in [9], [10], [11], [15], [16], [17], [26]. Since the model considered in this paper contains delay and the domain is non-smooth, our results are a further extension of [3], [27], [28] and [39]. (b) Denoting by P(X) the family of all nonempty subsets of X and considering a family of nonempty sets D 0 = {D 0 (t) : t ∈ R} ⊂ P(X), we give some preliminary definitions and theorems on the tempered pullback dynamics theory in what follows, see [6], [26].…”

“…For the Lipschitz-like case, motivated by [27,28] and based on estimates on the Stokes problem in [13] and [32], Brown, Perry and Shen [3] introduced the background flow in Lipschitz-like domains and proved the existence of the finite (fractal) dimensional universal attractor. Using the theory of pullback attractors, Yang, Qin, Lu and Ma [39] deduced the existence and regularity of pullback attractors, based on the background flow in Lipschitz-like domains. Inspired by [3], [27], [28], [37], [39] and [40], using the background flow ψ in Lipschitz-like domains which satisfies…”

“…Using the theory of pullback attractors, Yang, Qin, Lu and Ma [39] deduced the existence and regularity of pullback attractors, based on the background flow in Lipschitz-like domains. Inspired by [3], [27], [28], [37], [39] and [40], using the background flow ψ in Lipschitz-like domains which satisfies…”

Pullback dynamics for the 3‐D incompressible Navier–Stokes equations with damping and delay

“…Since J n are nonincreasing, we have that for all n ≥ n(k ) Using the energy estimates ( 27) and ( 28), the convergences (25) and Claim 1, we can deduce that ||u n (s)|| L 2 (t−h,t;V) → ||u(s)|| L 2 (t−h,t;V) .…”

“…Motivated by the recent results, [22][23][24][25] we will investigate the nonautonomous functional Navier-Stokes model with damping in a bounded domain Ω ⊂ R 3 which has sufficiently smooth boundary Ω; the problem for governing equations can be written as…”