Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

Foiaş, Ciprian; Jolly, Michael S.; Lan, Ruomeng; Rupam, Rishika; Yang, Yitao; Zhang, Bingsheng

doi:10.48550/arxiv.1312.0929

Cited by 2 publications

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“…Let U be the strong solution of (20) given by Theorem 2.7, and let R h be an interpolation operator satisfying either ( 6) or (7). Suppose the only knowledge we have about U is from the noisy observational measurements R h (U(t)) + ξ(t), that have been continuously recorded for times t ∈ [0, T ].…”

Section: The Deterministic Navier-stokes Equationsmentioning

confidence: 99%

“…Theorem 3.1. Suppose U is the strong solution of (20) given by Theorem 2.7, where U 0 ∈ V and f ∈ H. Moreover, assume R h : [ Ḣ1 ] 2 → [ L2 ] 2 satisfies (6) and that 2µc 1 h 2 ≤ ν. Then for any u 0 ∈ H and T > 0, there exists a unique stochastic process solution u of equation (21) in the following sense: P-a.s.…”

Section: The Deterministic Navier-stokes Equationsmentioning

confidence: 99%

“…We commence with the proof of Theorem 4.5. Assume that U is a strong solution of (20), that R h satisfies assumption (7) and that W is a [ Ḣ1 ] 2 -valued Q-Brownian motion. Assume that µ is large enough and that h is small enough such that µ ≥ 2νλ 1 GJ, and ν ≥ 2c 3 h 2 µ where…”

Section: Observations Of Nodal Valuesmentioning

confidence: 99%

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Continuous data assimilation with stochastically noisy data

2015

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We analyze the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimension incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the dataassimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the the limit supremum, as the time tends to infinity, of the expected value of the L 2 -norm and of the H 1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

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Section: The Deterministic Navier-stokes Equationsmentioning

confidence: 99%

Section: The Deterministic Navier-stokes Equationsmentioning

confidence: 99%

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Continuous data assimilation with stochastically noisy data

2015

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“…To make our presentation more self-contained, we include some of the relevant results from [6]. The first result gives some consequences of zero belonging to the attractor.…”

Section: Specific Preliminariesmentioning

confidence: 99%

“…For the sake of completeness, explicit expressions for R1 , R2 , R3 , C(g), β 1 , β 2 , δ 1 , δ 2 , δ 3 are recalled in the Appendix. The next result from [6] merely states a simple hierarchy of the spaces C(σ), σ ∈ R + . Proposition 3.3.…”

Section: Specific Preliminariesmentioning

confidence: 99%

On whether zero is in the global attractor of the 2D Navier–Stokes equations

et al. 2014

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Abstract. The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of C ∞ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.

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Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

Cited by 2 publications

References 0 publications

Continuous data assimilation with stochastically noisy data

Continuous data assimilation with stochastically noisy data

On whether zero is in the global attractor of the 2D Navier–Stokes equations

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