Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

Wang, Fengling; Li, Jia; Li, Yangrong

doi:10.1186/s13662-019-2165-6

Adv Differ Equ

2019

DOI: 10.1186/s13662-019-2165-6

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Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

Fengling Wang

Jia Li

Yangrong Li

Abstract: We consider a Wong-Zakai process, which is the difference of a Wiener-like process. We then prove that there are random attractors for non-autonomous Ginzburg-Landau equations driven by linear multiplicative noise in terms of Wong-Zakai process and Wiener-like process, respectively. Moreover, we establish the upper semi-continuity of random attractors as the size of difference noise tends to zero.

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Cited by 5 publications

(2 citation statements)

References 45 publications

(43 reference statements)

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“…Remark 2. The (complete) convergence of solutions holds true for reaction-diffusion equation (see [19,21,28,40]) and 1D-GL equations (see [22,38]). For the 2D-GL equation, since the Sobolev exponent in the 2D-Agmon inequality must be strictly larger than one, we need the higher regularity of initial data (e.g.…”

mentioning

confidence: 90%

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Li¹,

Wang²,

Yang³

2021

DCDS-B

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We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al. (Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zeromemory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory.

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mentioning

confidence: 90%

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Li¹,

Wang²,

Yang³

2021

DCDS-B

Self Cite

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“…In the case of linear noise, i.e. h is u or independent of u, the upper semi-continuity of A λ (ω) as λ → 0 is available, which follows from the same idea by Lu and Wang [26] (see also [11,34,37,39,41]) for other equations. But the upper semi-continuity of A λ (ω) as λ → 0 in the case of nonlinear noise remains open.…”

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confidence: 98%

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Yang

Long

2022

DCDS-B

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<p style='text-indent:20px;'>We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset <inline-formula><tex-math id="M1">\begin{document}$ \Lambda^* $\end{document}</tex-math></inline-formula> of the parameterized space such that the binary map <inline-formula><tex-math id="M2">\begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document}</tex-math></inline-formula> is continuous at all points of <inline-formula><tex-math id="M3">\begin{document}$ \Lambda^*\times \mathbb{R} $\end{document}</tex-math></inline-formula> with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space <inline-formula><tex-math id="M4">\begin{document}$ (0, \infty] $\end{document}</tex-math></inline-formula> and the infinity of noise means that the equation is deterministic.</p>

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Binary robustness of random attractors for 2D-Ginzburg–Landau equations with Wong–Zakai noise

Li,

Wang

2024

Stoch. Dyn.

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Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.

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Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

Cited by 5 publications

References 45 publications

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Binary robustness of random attractors for 2D-Ginzburg–Landau equations with Wong–Zakai noise

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