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:10.1093/imamat/hxu014

Cited by 19 publications

(22 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Similar bounds may be found in Temam [20] and Constantin and Foias [9]. The best estimate of ρ A to date appears in [14]. Before considering the case when f depends on time, we further note when f ∈ V is time independent that the bounds in (2.10) are finite for t > t 0 .…”

Section: Preliminariessupporting

confidence: 67%

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Çelik¹,

Olson²,

Titi³

2019

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.and let Q λ = I − P λ be the orthogonal complement of P λ . Now, given λ > 0 and I h define J = P λ P σ I h and E = I − J.(1.6) Note, although no additional orthogonality or regularity properties other than those appearing in Definition 1.1 have been assumed on I h , the above spectral filtering yields an operator J which is nearly orthogonal and has a range contained in D(A). The downscaling data assimilation algorithm studied in this paper may now be stated as Definition 1.2. Let U be an exact solution of (1.2) which evolves according to dynamics given by the semi-process S. Let t n = t 0 + nδ be a sequence of times for which partial

show abstract

Section: Preliminariessupporting

confidence: 67%

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Çelik¹,

Olson²,

Titi³

2019

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…Therefore, τ t0 u ∈ T + b . The bound in (4.11) is easily seen by taking the inner product of (2.3) in H with Au and performing the usual estimates, while the bound in (4.12) follows analogously to the proof of (2.15) in [31].…”

Section: ) Andmentioning

confidence: 90%

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations

Biswas

Foiaş

Mondaini

et al. 2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier-Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier-Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier-Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.

show abstract

“…Remark 5.1. Using the results from the paper [5], one could show further that the convergence in case (ii) is also valid for any higher norms.…”

Section: A Convergence Propertymentioning

confidence: 95%

On a Subclass of Solutions of the 2D Navier–Stokes Equations with Constant Energy and Enstrophy

Tian

You

2019

J Dyn Diff Equat

View full text Add to dashboard Cite

It is not yet known if the global attractor of the space periodic 2D Navier-Stokes equations contains nonstationary solutions u(x, t) such that their energy and enstrophy per unit mass are constant for every t ∈ (−∞, ∞). The study of the properties of such solutions was initiated in [2], where, due to the hypothetical existence of such solutions, they were called "ghost solutions". In this work, we introduce and study geometric structures shared by all ghost solutions. This study led us to consider a subclass of ghost solutions for which those geometric structures have a supplementary stability property. In particular, we show that the wave vectors of the active modes of this subclass of ghost solutions must satisfy certain supplementary constraints. We also found a computational way to check for the existence of these ghost solutions.

show abstract

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

Cited by 19 publications

References 16 publications

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations

On a Subclass of Solutions of the 2D Navier–Stokes Equations with Constant Energy and Enstrophy

Contact Info

Product

Resources

About