Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation

Li, Yangrong; Yang, Su‐Jau; Zhang, Qiangheng

doi:10.3934/era.2020080

Cited by 6 publications

(5 citation statements)

References 28 publications

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“…We write the weak solution of (3.8) as 35), where (•) h is the mollification operator discussed above, for 0 ≤ t < T , we get…”

Section: Rds Generated By 2d Scbf Equationsmentioning

confidence: 99%

“…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%

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$\mathbb{H}^1$-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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“…We write the weak solution of (3.8) as 35), where (•) h is the mollification operator discussed above, for 0 ≤ t < T , we get…”

Section: Rds Generated By 2d Scbf Equationsmentioning

confidence: 99%

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 KS equation was introduced in [13,16], which describes turbulence phenomena in chemistry and combustion. The well-posedness and dynamics of KS equations without delay have been studied in [15,17] for the deterministic case and in [12,14,22,23] for the stochastic case. To the best of our knowledge, the long-term behavior of delay KS equations has not been considered.…”

Section: Introductionmentioning

confidence: 99%

Regular dynamics of stochastic nondissipative retarded Kuramoto–Sivashinsky equations

Zhang

2023

Mathematische Nachrichten

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The theme of this paper is to study the long‐term behavior of stochastic nondissipative Kuramoto–Sivashinsky (KS) equations with time‐dependent force and abstract delays. We first introduce an odd function to overcome the nondissipation of this equation. We then use the spectrum decomposition technique to prove the existence and regularity of a unique pullback random attractor (PRA), and establish the measurability, forward compactness, and long‐time stability of PRAs in the regular space. Finally, we consider the upper semicontinuity of PRAs in the regular space from nonautonomy to autonomy. Moreover, we provide three concrete examples for the general delay term.

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“…Note that Wong-Zakai approximation systems in these references were only considered in the case of linear noise, that is, the sequence of diffusion functions G = (G i ) i∈Z in ( 1) is u = (u i ) i∈Z or independent of u. Moreover, so were all the results on the semicontinuity of random attractors, see 3,22,37,43 for autonomous stochastic equations, and 4,7,20,21,23,26,39,42,45,47 for non-autonomous stochastic equations.…”

Section: Introductionmentioning

confidence: 99%

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise

Yang

Caraballo

2022

Journal of Mathematical Physics

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In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tail-estimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved.

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Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation

Cited by 6 publications

References 28 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

Regular dynamics of stochastic nondissipative retarded Kuramoto–Sivashinsky equations

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise

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