Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise

Li, Yangrong; Yang, Su‐Jau

doi:10.3934/cpaa.2019056

Cited by 15 publications

(11 citation statements)

References 35 publications

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“…This means φ is backward asymptotically compact in V g (O) (see [12]). Therefore, our result follows from an abstract result on bi-spatial random attractor given in [15] and [17] immediately.…”

supporting

confidence: 59%

“…Some abstract theoretical criteria were established in [6,16,25,26] for deterministic cases and [17] for stochastic ones. In order to establish the existence result of a backward compact random attractor in H g (O), where H g (O) is a special subspace of the weighted space L 2 g (O) 2 , we assume that the force f is backward tempered and the so-called backward asymptotic compactness, which indicates that the usual asymptotic compactness is uniform in the past, is introduced.…”

Section: 3)mentioning

confidence: 99%

“…On the other hand, we know that φ is D-backward asymptotically compact. Then it follows from [17] that φ has a D-pullback backward compact random attractor. 6.…”

mentioning

confidence: 99%

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Regular measurable backward compact random attractor for <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes equation

Li¹,

Xu²,

Yu³

2020

Communications on Pure &Amp; Applied Analysis

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In this paper, we study the backward compactness of random attractors, which describes the compactness of the union ∪ s≤τ A(s, ω) of random attractor sections over past times, τ ∈ R. In particular, we prove the backward compactness and the regularity of random attractors for stochastic g-Navier-Stokes equations under the condition that the force is backward tempered and backward limiting. The attraction universe in consideration is non-autonomous and consists of backward tempered sets.

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supporting

confidence: 59%

Section: 3)mentioning

confidence: 99%

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Regular measurable backward compact random attractor for <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes equation

Li¹,

Xu²,

Yu³

2020

Communications on Pure &Amp; Applied Analysis

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“…A schematic illustration of the dislocation mechanisms operative in different length scale regimes is given in Figure . With decreasing layer thickness starting in the micron range the dominant deformation mechanisms change from dislocation pile‐up to confined layer slip (CLS) of dislocations (<50 nm) and finally to interface crossing of dislocations (slip transfer) (<2.5 nm) leading to strain localization that limits the uniform deformability . These mechanisms may be transferred to ARB by considering the peculiarities of interface‐mediated plasticity.…”

Section: Selected Arb Systemsmentioning

confidence: 99%

Microstructure, Texture, and Mechanical Properties of Laminar Metal Composites Produced by Accumulative Roll Bonding

et al. 2018

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The paper aims to summarize the research on Laminar Metal Composites produced by Accumulative Roll Bonding. After some notes on the general subject, frequent material combinations and the issue of bonding are addressed. Then, the evolution of microstructure, texture, and mechanical properties typically observed in such materials is briefly summarized. Furthermore, the crucial aspect of layer continuity is discussed. In the main part, detailed experimental insight is provided for three representatives of different structural developments, namely Al2N/Al5N, Cu/Nb, and Ti/Al. The chosen systems represent different levels of component dissimilarity, as Al2N/Al5N is a combination of the same face centered cubic metal with varying purity while Cu/Nb and Ti/Al are combinations of face centered cubic with both body centered cubic and hexagonal close packed metals, respectively. As the layer thicknesses span different ranges, the composites also illustrate both the opportunities and experimental challenges of ARB Laminar Metal Composites.bonding (in the following, those processes will be referred to "ARB"), often supplemented by annealing and rolling steps. Naturally, a high percentage of the scientific attention regarding LMCs is bestowed on Al alloys, for example, [7,25,[43][44][45][46][47][48][49][50] whose assets are excellent formability and corrosion resistance while being low in both weight and price.Al alloys are frequently combined in order to improve specific properties, for example, [51][52][53][54][55] A strong but corrosion susceptible Al alloy of series 2xxx, for example, benefits from the combination with a less strong but corrosion resistant alloy of series 6xxx. [50] Likewise, in LMCs of series 5xxx and 6xxx, both surface quality and weldability are improved due to the reduction of the PLC-effect [53] and hot cracking, [55] respectively.By combining Al with both Fe [56][57][58] or Cu, [45,59] the magnetic properties of the former and the electrical properties of the latter can be used to produce LMCs with potential in microelectronics and in electrical power industry, [60] respectively. Other components in LMCs with Cu are Ag, [61] Zn, [62] both Zn and Al, [63] and both Al and Ni. [64] Nb receives major scientific attention due to its superconducting properties, which work especially well in a finelayered structure with Cu or Al, [65] which are chosen for their low solubility in Nb and the absence of mutual intermetallics. In contrast, other LMCs are produced especially for intermetallic transition, for instance Al/Fe, [57,58,66] Al/Mg, [43,44] Al/Ti, [67][68][69][70][71] Al/Ni, [72,73] Ti/Nb, [74] or Ti/Al/Nb. [74] When high specific strength in combination with sufficient formability and ductility is the foremost requirement, Al is often combined with high-strength partners such as steel, for example, [66,75] Mg, for example, [46,76,49] Ti, for example, [67][68][69]74,[77][78][79][80][81][82][83][84][85][86] as well as both Mg and Ti. [87] Furthermore, Ti is paired with Fe, [88,89] Cu, for example, [90,...

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“…We must prove that the random dynamical system (or cocycle) is backward asymptotically compact in X , which means that the pullback asymptotic compactness is uniform in the past. By combining the bounded backward absorption, we need to establish the backward compactness [10][11][12][13][14] of the PRA  , which means that ∪ s≤  (s, ) is relatively compact in X . The backward compactness in X is available via the Ascoli-Arzelà theorem and it often leads to the longtime stability of the PRA in X as in (1).…”

Section: Introductionmentioning

confidence: 99%

Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

Zhang

2020

Math Methods in App Sciences

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We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets Aϱ(t,·), where t is the current time and ϱ is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero‐memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous p‐Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time‐dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.

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Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise

Cited by 15 publications

References 35 publications

Regular measurable backward compact random attractor for <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes equation

Regular measurable backward compact random attractor for <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes equation

Microstructure, Texture, and Mechanical Properties of Laminar Metal Composites Produced by Accumulative Roll Bonding

Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

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