Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains

In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincaré domains (which may be bounded or unbounded). Our aim is to investigate the existence and AAR of random attractors for 3D SGMNSE when the time-dependent forcing converges to a time-independent function under the perturbation of linear multiplicative noise as well as additive noise. The main approach is to provide a way to justify that, on some uniformly tempered universe, the usual pullback asymptotic compactness of the solution operators is uniform across an infinite time-interval $(-\infty,τ]$. The backward uniform tail-smallness'' and flattening-property’’ of the solutions over $(-\infty,τ]$ have been demonstrated to achieve this goal. To the best of our knowledge, this is the first attempt to establish the existence as well as AAR of random attractors for 3D SGMNSE on unbounded domains.

💡 Research Summary

The paper investigates the long‑time stochastic dynamics of three‑dimensional globally modified Navier‑Stokes equations (SGMNSE) on domains that satisfy a Poincaré inequality, which may be bounded or unbounded. The authors consider both additive and linear multiplicative noise and a time‑dependent external force that converges, as time goes to infinity, to a stationary forcing. Their main objectives are (i) to prove the existence of a pullback random attractor for the non‑autonomous stochastic system and (ii) to establish asymptotic autonomous robustness (AAR): the random attractor of the non‑autonomous system converges almost surely to the random attractor of the associated autonomous system when the forcing becomes time‑independent.

To overcome the lack of compact Sobolev embeddings on unbounded domains, the authors adopt a novel approach based on Kuratowski’s measure of non‑compactness. They show that uniform pullback asymptotic compactness can be achieved on an infinite time interval ((-\infty,τ]) provided two uniform properties hold: (a) backward tail‑smallness, i.e., the energy of solutions outside large balls becomes arbitrarily small uniformly in time, and (b) a flattening property, i.e., high‑order Sobolev norms can be uniformly approximated by low‑order norms on bounded subdomains. Lemma 2.12 formalises this idea and serves as the key technical tool.

The paper first reformulates the stochastic system in an abstract functional‑analytic setting. The velocity field lives in the divergence‑free space (V) and its (L^2) closure (H). The Stokes operator (A) and the Helmholtz projection (P) are introduced, together with the modified nonlinear operator (B_N(u)=F_N(|u|_V)P(u·∇u)), where (F_N) is a cut‑off function that prevents blow‑up by damping the nonlinearity when (|u|_V) is large. The pressure term is recovered via the invertibility of the Laplacian on Poincaré domains, yielding an explicit representation (1.7) that can be estimated in (L^2).

Two main theorems are proved. Theorem 2.8 treats the case of additive noise, establishing the existence of a pullback random attractor (\mathcal{A}τ(ω)) and its convergence to the autonomous attractor (\mathcal{A}∞(ω)) as the forcing stabilises. Theorem 2.10 handles linear multiplicative noise, with essentially the same conclusions. The proofs consist of several steps:

1. Uniform absorbing sets on a uniformly tempered universe are constructed; their radii involve supremums over the infinite past, but are shown to be finite using energy estimates and the properties of the Ornstein‑Uhlenbeck process that models the stochastic forcing.

2. Uniform pullback asymptotic compactness is obtained by verifying the tail‑smallness (Lemma 3.3) and flattening (Lemma 3.7) estimates for the solution trajectories over ((-\infty,τ]). These estimates are uniform with respect to the tempered universe, which is crucial for handling the infinite time interval.

3. Kuratowski measure is applied (Lemma 2.12) to translate the uniform tail‑smallness and flattening into a vanishing non‑compactness measure, guaranteeing pre‑compactness of the union of pullback images of any bounded set.

4. Measurability of the attractor is addressed. Because the absorbing sets involve supremums over an uncountable past, the authors prove that the uniformly pullback compact attractor coincides with the usual pullback random attractor, thereby inheriting its measurability (Step VI in Section 3.5).

5. Asymptotic autonomous robustness follows from the uniform compactness and the convergence of the forcing term. By comparing the non‑autonomous cocycle with the autonomous one and using the uniform estimates, the Hausdorff distance between (\mathcal{A}τ(ω)) and (\mathcal{A}∞(ω)) is shown to tend to zero almost surely, establishing (1.6).

The paper also includes an appendix where a different modification, based on an (L^4) norm cut‑off, is introduced. Existence and uniqueness of weak and strong solutions for this variant are proved (Theorems A.2 and A.3), indicating that the analytical framework can be extended to other globally modified models.

Overall, the work makes a significant contribution by being the first to prove both the existence of random attractors and their asymptotic autonomous robustness for three‑dimensional globally modified Navier‑Stokes equations on unbounded Poincaré domains. The combination of tail‑smallness, flattening, and Kuratowski’s non‑compactness measure provides a robust methodology that may be applicable to a broad class of infinite‑dimensional stochastic partial differential equations lacking compact embeddings.