On the domain of analyticity and small scales for the solutions of the damped-driven 2d Navier–Stokes equations

Ilyin, Alexei; Titi, Edriss S.

doi:10.4310/dpde.2007.v4.n2.a1

Cited by 3 publications

(4 citation statements)

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“…It suffices to consider I 5 and I 6 . Indeed, observe that I 2 = 0, while the terms I 1 − I 4 will be estimated exactly as in [35]. Collecting these estimates, we arrive at…”

Section: )mentioning

confidence: 97%

“…In the next section, we will derive uniform bounds for Ψ n and show that these uniform bounds ensure that {η (n) (• , t) + iΘ (n) (• , t)} n≥0 is normal family. It will then follow as in [20,35] that, upon possibly passing to a subsequence, η (n) + iΘ (n) → η + iΘ, satisfying (4.9), which satisfies certain lower bounds on its analyticity radius.…”

Section: )mentioning

confidence: 99%

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Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations

Biswas

Martinez

2017

Nonlinear Analysis: Real World Applications

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We consider the two-dimensional (2D) Navier-Stokes equations (NSE) with space periodic boundary conditions and an algorithm for continuous data assimilation developed by Azouani, Olson and Titi (2014). The algorithm is based on the observation that existence of finite determining parameters for nonlinear dissipative systems can be exploited as a feedback control mechanism for a companion system into which observables, e.g, modes, nodes, or volume elements, are input directly for the purpose of assimilation. It has been shown that in the case of the 2D NSE, the approximating solution induced by the algorithm synchronizes with the exact solution of the 2D NSE in the topology of the Sobolev space, H 1 , provided that the number of observed modes, nodes, or volume elements is sufficiently large in terms of the Grashof number. In this article, we adapt a technique, introduced by Grujić and Kukavica (1998) and Kukavica (1999) to obtain good estimates of the analyticity radius for the 2D NSE, and show that one can in fact obtain synchronization in the analytic Gevrey class in the case of modal observables, given sufficiently many, but fixed number of such observables. For these types observables, we additionally show that synchronization in the uniform norm, L ∞ , can be achieved by assuming the same number of modal observables (in terms of the order of the Grashof number) as is required for the H 1 synchronization.

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“…It suffices to consider I 5 and I 6 . Indeed, observe that I 2 = 0, while the terms I 1 − I 4 will be estimated exactly as in [35]. Collecting these estimates, we arrive at…”

Section: )mentioning

confidence: 97%

Section: )mentioning

confidence: 99%

Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations

Biswas

Martinez

2017

Nonlinear Analysis: Real World Applications

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“…Proof. Recall that v(t) satisfies equation (11), v (2) (t) satisfies (14). In general, v (m) (t), for m > 2, satisfies equation (19).…”

Section: Asymptotic Approximation In V Mmentioning

confidence: 99%

“…In this section we would like to derive a lower bound for the ℓ NSV , similar to relation (45) for the 3D Navier-Stokes equations. For other estimates on a related smallest length scale (via computation of the radius of analyticity of the solutions) of the Navier-Stokes equations in 2 and 3 dimensions see [12], [13], and [19] (see also [14]). See also [6] and [9] for other approach to this subject.…”

Section: Estimating the Exponential Decaying Small Scalementioning

confidence: 99%

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

2008

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Recently, the Navier-Stokes-Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides an additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier-Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale -the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier-Stokes equations. Our estimate coincides with similar available lower bound for the smallest dissipation length scale of solutions of the 3D Navier-Stokes equations.MSC Classification: 35Q30, 35Q35, 35B40, 35B41, 76F20, 76F55

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