Large-domain stability of random attractors for stochastic g-Navier–Stokes equations with additive noise

Li, Fuzhi; Xu, Dongmei; She, Lianbing

doi:10.1186/s13660-020-02459-w

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…The concept of an asymptotically compact cocycle was introduced in [14] and the authors proved the existence of attractors fornon-autonomous 2D NSE. Later, several authors used this method to prove the existence of random attractors in unbounded domains, see for example [9,12,13,27,35,36,38,47,49,53,54,58] etc. The existence of a unique random attractor for the 2D and 3D SCBF equations (1.2) perturbed by additive rough noise in H is proved in [38].…”

Section: Introductionmentioning

confidence: 99%

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

Kinra¹,

Mohan²

2021

Preprint

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The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincaré domains). We first prove the existence of H 1 -random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent r ∈ [1, 3]) perturbed by an additive noise on Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in H 1 for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these results to general unbounded domains is also discussed. Finally, for 2D SCBF equations forced by additive one-dimensional Wiener noise, we prove the upper semicontinuity of the random attractors, when the domain changes from bounded to unbounded (Poincaré).

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Section: Introductionmentioning

confidence: 99%

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

Kinra¹,

Mohan²

2021

Preprint

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“…The concept of an asymptotically compact cocycle was introduced in [11] and the authors proved the existence of attractors for non-autonomous 2D NSE. Later, several authors used this method to prove the existence of random attractors on unbounded domains, see for example [5,6,27,33,51] etc. Two of the methods used to prove asymptotic compactness without Sobolev embeddings are as follows: the method of energy equations (see [19,27,52] etc) and the method of uniform tail estimates (see [54,58] etc).…”

Section: Introductionmentioning

confidence: 99%

Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise

Kinra,

Mohan

2021

Preprint

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The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with nonlinear diffusion terms on bounded and unbounded (R d for d = 2, 3) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the literature, CBF equations are also known as Navier-Stokes equations (NSE) with damping, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE. The presence of linear damping term helps to establish the results in the whole domain R d . The nonlinear damping term supports to obtain better results in 3D and also for a large class of nonlinear diffusion terms. Moreover, we prove the existence of a unique random pullback attractor for stochastic CBF equations with additive white noise. Finally, for additive as well as multiplicative noise case, we establish the convergence of solutions and upper semicontinuity of random pullback attractors for Wong-Zakai approximations of stochastic CBF equations towards the random pullback attractors for stochastic CBF equations when correlation time of colored noise converges to zero.

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“…The concept of an asymptotically compact cocycle was introduced in [16] and authors have established the existence of attractors for the non-autonomous 2D Navier-Stokes equations. Later, this concept has been utilized to prove the existence of random attractors for several SPDEs like 1D stochastic lattice differential equation [5], stochastic Navier-Stokes equations on the 2D unit sphere [4], stochastic g-Navier-Stokes equations [27,36,39], stochastic nonautonomous Kuramoto-Sivashinsky equations [38], stochastic heat equations in materials with memory on thin domains [48], stochastic reaction-diffusion equations [6,47,51], 3D stochastic Benjamin-Bona-Mahony equations [51], etc and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains

Kinra¹,

Mohan²

2020

Preprint

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In this paper, we consider the two-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations ), in unbounded domains like Poincaré domains and discuss about the asymptotic behavior of its solution. We establish the existence of random attractors for the stochastic flow generated by the 2D SCBF equations for r ∈ [1, ∞) with additive rough noise on Poincaré domains. One of the technical difficulties connected with the rough noise is overcame with the help of the corresponding Cameron-Martin space (or Reproducing Kernel Hilbert space). Furthermore, we observe that the regularity of the rough noise needed to obtain random attractors for the 2D SCBF equations for r ∈ [1, 3] is same as that in the case of 2D Navier-Stokes equations, where as for the case r ∈ (3, ∞), we require more spatial regularity on the noise.

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Large-domain stability of random attractors for stochastic g-Navier–Stokes equations with additive noise

Cited by 4 publications

References 21 publications

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise

Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains

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