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

Bui Kim My, Ho Thi Hang, Kush Kinra, Manil T. Mohan, Pham Tri Nguyen

Abstract

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.

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

Abstract

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 . The backward uniform ``tail-smallness'' and ``flattening-property'' of the solutions over 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.
Paper Structure (24 sections, 24 theorems, 228 equations)

This paper contains 24 sections, 24 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Literature survey
  3. Aims, difficulties and approaches
  4. Outline
  5. Mathematical Formulations and Auxiliary Results
  6. Function spaces and operators
  7. Linear operator
  8. Nonlinear operator
  9. Abstract formulation and Ornstein-Uhlenbeck process
  10. Ornstein-Uhlenbeck process
  11. Backward-uniformly tempered random set
  12. Class of random sets
  13. Assumption and main results
  14. Kuratowski's measure of non-compactness
  15. Asymptotically Autonomous Robustness of Random Attractors for \ref{['1']}: Additive Noise
  16. ...and 9 more sections

Key Result

Lemma 2.4

For any $\boldsymbol{u},\boldsymbol{v}\in\mathbb{V}$ and each $N>0$,

Theorems & Definitions (51)

  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Romito_2009
  • Lemma 2.5
  • proof
  • Remark 2.7
  • Theorem 2.8: Additive noise case
  • Remark 2.9
  • ...and 41 more