Stability threshold of two-dimensional Couette flow in Sobolev spaces

Masmoudi, Nader; Wei, Zhao

doi:10.4171/aihpc/8

Cited by 29 publications

(14 citation statements)

References 35 publications

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“…The determination of the smallest γ for which nonlinear stability still holds is an important problem, and is an active research area, aiming to address the "transition threshold conjecture". We refer the reader to [2,12,13,14,19,20,22,36,39,40,51,55,56] and references therein for important recent works on enhanced dissipation and transition threshold problems, and section 6.11 of the recent book [6] for an excellent survey.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Surprisingly, results in this direction are very few, as pointed out in [6]. The only results addressing precise uniform inviscid damping, as far as we know, are [12,39,3,7] for the Couette flow with full nonlinear analysis or precise linear results, and [23] for the spectrally restricted stream function for "mixing layer" type shear flows (but a description of the full solution without spectral restrictions is not available).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case

Jia¹

2022

Preprint

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We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on T×R. The main task is to understand the associated Rayleigh and Orr-Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr-Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr-Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on T × R. The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere. Contents

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case

Jia¹

2022

Preprint

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“…Instead, one may hope to establish the existence of a nonlinear stability threshold depending on characteristic quantities of the flow (the so-called Reynolds number, here inversely proportional to the kinematic viscosity ν > 0): for initial data below the threshold, the nonlinear problem can be treated perturbatively and linear effects persist, whereas above it turbulent motion and instabilities may occur. And indeed, results of this type have been first demonstrated for monotone shear flows, with subsequent refinements on the precise size of the threshold [7,9,29,30] Our previous work [14] proved the existence of such a threshold near the Poiseuille flow in the Navier-Stokes equations, while for the bar states on rectangular tori T 2 δ with 0 < δ < 1 this was shown in [36]. In stark contrast to these results, we show here that the corresponding result cannot hold for the bar states on T 2 .…”

Section: Theorem 1 (Stationary States Near Kolmogorovmentioning

confidence: 88%

Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

Zelati¹,

Elgindi²,

Widmayer³

2020

Preprint

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We study the behavior of solutions to the incompressible 2d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems.We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus T 2 . This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an "inviscid damping" mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible.Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on T 2 . In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity H 3 resp. H 5+ .In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on T 2 . Contents

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“…Previous studies have investigated plane Couette flow with (x, y) ∈ T×R ( see [2,4,5,23,24]) and Couette flow in a finite channel with (x, y) ∈ T × [−1, 1] (see [3,7]). In these works, the spatial variable x corresponding to the non-shear direction was defined on a torus.…”

Section: Comparison With Couette Flow From An Operator Perspectivementioning

confidence: 99%

“…Significant advancements and findings related to the subcritical transition regime for the 2D incompressible NS equations have emerged recently. These developments include researches on Couette flow without boundaries [2,4,5,23,24] and with boundaries [3,7], on Lamb-Oseen vortices [12][13][14][15]21] and on periodic Kolmogorov flow [34,35]. Recently, Guo, Pausader and Widmayer [16] has made significant progress in the stability problem of the Euler flow with rotation.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Asymptotic Stability and Transition Threshold for 2D Taylor–Couette Flows in Sobolev Spaces

An,

He,

2024

Commun. Math. Phys.

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In this paper, we investigate the stability of the 2-dimensional (2D) Taylor–Couette (TC) flow for the incompressible Navier–Stokes equations. The explicit form of velocity for 2D TC flow is given by $$u=(Ar+\frac{B}{r})(-\sin \theta , \cos \theta )^T$$ u = ( A r + B r ) ( - sin θ , cos θ ) T with $$(r, \theta )\in [1, R]\times \mathbb {S}^1$$ ( r , θ ) ∈ [ 1 , R ] × S 1 being an annulus and A, B being constants. Here, A, B encode the rotational effect and R is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor–Couette flow is asymptotically stable, even at high Reynolds number ($$Re\sim \nu ^{-1}$$ R e ∼ ν - 1 ), with a sharp exponential decay rate of $$\exp (-\nu ^{\frac{1}{3}}|B|^{\frac{2}{3}}R^{-2}t)$$ exp ( - ν 1 3 | B | 2 3 R - 2 t ) as long as the initial perturbation is less than or equal to $$\nu ^\frac{1}{2} |B|^{\frac{1}{2}}R^{-2}$$ ν 1 2 | B | 1 2 R - 2 in Sobolev space. The powers of $$\nu $$ ν and B in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor–Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient B. The larger the |B|, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor–Couette flows.

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Stability threshold of two-dimensional Couette flow in Sobolev spaces

Cited by 29 publications

References 35 publications

Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case

Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case

Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

Nonlinear Asymptotic Stability and Transition Threshold for 2D Taylor–Couette Flows in Sobolev Spaces

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