Stability results for 2D Navier–Stokes equations with unbounded delay

Liu, Linfang; Caraballo, Tomás; Marín-Rubio, Pedro

doi:10.1016/j.jde.2018.07.008

Cited by 37 publications

(25 citation statements)

References 32 publications

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“…If the material in fluid flow is special, then the governing equations becomes 2D incompressible Navier‐Stokes system with delay: continuous or distributed cases. For the well‐posedness and dynamic systems for 2D Navier–Stokes flow with delay, we can refer to other studies 17‐28 and some more generalized fluid flow model with delay in Li et al 29 and Lukaszewicz and Kali 30 . The pullback dynamics for the 2D Navier–Stokes flow has been presented in above literatures, but the fractal dimension and robustness of pullback attractors have not been solved till now.…”

Section: Introductionmentioning

confidence: 99%

“…we can refer to other studies [17][18][19][20][21][22][23][24][25][26][27][28] and some more generalized fluid flow model with delay in Li et al 29 and Lukaszewicz and Kali. 30 The pullback dynamics for the 2D Navier-Stokes flow has been presented in above literatures, but the fractal dimension and robustness of pullback attractors have not been solved till now.…”

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confidence: 99%

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The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

Yang

Guo

et al. 2020

Math Methods in App Sciences

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

Yang

Guo

et al. 2020

Math Methods in App Sciences

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“…Given u : (−∞, T ) → X, for each t ∈ (0, T ), we define the function u t on τ ∈ (−∞, 0]. Referring to [4,27], we now make the following assumptions:…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

Zheng¹,

Liu²,

Liu³

2019

Preprint

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We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property. stands for memory effect whose kernel κ is a nonnegative summable function of total mass ∞ 0 κ(s)ds = 1 having the explicit form κ(s) = ∞ s µ(r)dr. The Navier-Stokes equation reflects the basic mechanical law of viscous fluid flow, thus are very significant both in a purely mathematical sense and in the fluid applications including physics and biology. Since the first relative paper of Leray [24] published in 1933, the Navier-Stokes equation has been the object of numerous works. Caraballo and Han [12] considered the Navier-Stokes equations with delay effect which shows not only the present state but also its history in both autonomous and non-autonomous cases (see [15,31,39] and references therein). In 2001, Caraballo and Real [13] put forward the possibility of some kind of delay appearing in the Navier-Stokes equations in an open and bounded domain Ω ⊂ R N (N = 2 or 3) with regular boundary Γ:(1.2) and they proved the existence and uniqueness of solutions. Subsequently, they established the existence of a pullback attractor when N = 2 in [14]. Concerning the Navier-Stokes equation with finite delay in an unbounded domain, Garrido-Atienza and Marín-Rubio [20] addressed the existence and uniqueness of solutions for both the evolutionary and the stationary cases. Especially, in the three-dimensional case, they only proved the existence of solutions. These results were extended in R N with 2 ≤ N ≤ 4 by Niche and Planas [37]. For unbounded delay, the authors investigated the well-posedness and asymptotic behavior of solutions [23, 30, 35] and their references. In addition, Liu, Caraballo and Marín-Rubio [27]studied the existence of pullback attractors for two-dimensional Navier-Stokes equation with infinite delay by using the phase space BCL −∞ (H) which will be defined below rather than C γ (H) carried out in [31] for a 2D Navier-Stokes model with infinite delay due to lacking the term ∆(∂ t u). Recently, Anh and Thanh [4] discussed the 3D Navier-Stokes-Voigt equations with infinite delay of the formwhere α is length-scale parameter describing the elasticity of the fluid. Some results on the existence and uniqueness of weak solutions and the existence of global attractors were proved. Other types of delay including constant, bounded variable delay as well as bounded distributed delay can be found in [10,13] and reference therein.As for memory term, it plays an important role in the description of several phenomena such as non-Newtonian flows, soil mechanics and heat conduction theory and prescribes the effect of past history and provides a more realistic de...

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“…By (27), there is T > 0 such that for all r ≤ −T , which implies that for all t ≥ T and r ≤ 0, e − −t r−t γρ(θrω)dr e −2bz(θr−tω) = e 0 −t − 0 r−t γρ(θrω)dr e −2bz(θr−tω) =e 0 −t (γρ(θrω)−γρ)dr+γρt e − 0 r−t (γρ(θrω)−γρ)dr−γρ(t−r) e −2bz(θr−tω) ≤e 4 t+γρt+ 4 (t−r)−γρ(t−r) e 4 (t−r) = e 3 4 t e (γρ− 2 )r .…”

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confidence: 99%

“…Thereafter, many scholars (e.g. in [2,5,13,15,19,26,27,41,39,42]) have studied dynamics in terms of attractors for PDEs with delays.…”

mentioning

confidence: 99%

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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Stability results for 2D Navier–Stokes equations with unbounded delay

Cited by 37 publications

References 32 publications

The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

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

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