Attractors for a double time-delayed 2D-Navier-Stokes model

García-Luengo, Julia; Marín-Rubio, Pedro; Planas, Gabriela

doi:10.3934/dcds.2014.34.4085

Cited by 25 publications

(38 citation statements)

References 26 publications

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“…Now we provide some examples of (unbounded) delay forcing terms which can be set within our general set-up (see [22,19,20,18,21]).…”

Section: Preliminariesmentioning

confidence: 99%

“…The analysis of the Navier-Stokes equations with hereditary terms was initiated by Caraballo and Real in [14], and developed in [11,12,13,7,15,8,9,10,5,6], where several issues have been investigated: the existence and uniqueness of solution, stationary solution, the existence of attractors (global, pullback and random ones) and the local exponential stability of state-steady solution of Navier-Stokes models with several types of delay (constant, bounded variable delay as well as bounded distributed delay). In the papers [22,33,29,34,30,31,35,19,32,20,18,21] the authors have discussed the asymptotic behavior and regularity of solutions of 2D Navier-Stokes equations (and 3D-variations of Navier-Stokes models) with delay (finite and infinite). Wei and Zhang [39] have obtained the exponential stability and almost sure exponential stability of the weak solution for stochastic 2D Navier-Stokes equations with bounded variable delays by using the approach proposed in [14,6].…”

Section: Introductionmentioning

confidence: 99%

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

Liu¹,

Caraballo²,

Marín-Rubio³

2018

Journal of Differential Equations

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Some results related to 2D Navier-Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax-Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.

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“…Now we provide some examples of (unbounded) delay forcing terms which can be set within our general set-up (see [22,19,20,18,21]).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Liu¹,

Caraballo²,

Marín-Rubio³

2018

Journal of Differential Equations

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“…Many other works have been done later on, providing more information on the regularity of such an attractor (e.g., [24], [23]). Also there are other works dealing with the cases in which the delay appears not only in the forcing terms but also in the convective and diffusion terms (see [3], [25], [21]) and at the same time, there exist also several papers in which the delay is allowed to be unbounded (see [37], [22]).…”

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confidence: 99%

A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Caraballo¹,

Han²

2015

Discrete &Amp; Continuous Dynamical Systems - S

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In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.

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“…Then Planas and collaborators [17,10,11] treat problem (1), the analysis of well-posedness (including uniqueness) and an unbounded delay case too. The asymptotic behavior in the sense of attractors in L 2 -norm is carried out in [4]. It is worth also to mention that in [20] the inclusion of a delay is used as an approximation to a 3D Navier-Stokes model when the length of the delay vanishes.…”

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confidence: 99%

“…It is worth also to mention that in [20] the inclusion of a delay is used as an approximation to a 3D Navier-Stokes model when the length of the delay vanishes. Nevertheless, in the 2D case our interest in the problem is just mathematical, due to the difficulties arising in controlling the norm of the derivatives as cited in the abstract (see also [4]). Our goal in this paper is to improve the results in [4] providing regularity for both solutions and attractors.…”

mentioning

confidence: 99%

Some regularity results for a double time-delayed 2D-Navier-Stokes model

García-Luengo¹,

Marín-Rubio²,

Planas³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model-from the mathematical point of view-is that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the L 2-framework obtained by the authors [Discrete Contin. Dyn. Syst. 34 (2014), 4085-4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided.

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Attractors for a double time-delayed 2D-Navier-Stokes model

Cited by 25 publications

References 26 publications

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

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

A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Some regularity results for a double time-delayed 2D-Navier-Stokes model

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