On a larger class of stable solutions to the Navier-Stokes equations in exterior domains

Kozono, Hideo; Yamazaki, Masao

doi:10.1007/pl00004644

Cited by 61 publications

(64 citation statements)

References 31 publications

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“…This idea motivated to study the exterior problem for the incompressible Navier-Stokes system in the space L 3,∞ (Ω) (see e.g. [4,18,19,28,31] and the references given there). Results in this direction were also obtained in the recent paper [3], where ideas from Theorem 2.2 were adapted.…”

Section: Theorem 24 Denote By U(x T) and W(x T) The Solutions To Tmentioning

confidence: 99%

About the Regularized Navier?Stokes Equations

Cannone

Karch

2005

J. math. fluid mech.

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The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes system. The Marcinkiewicz space L 3,∞ is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical "regularized" Navier-Stokes systems. The first one was introduced by J. Leray and consists in "mollifying" the nonlinearity. The second one was proposed by J.L. Lions, who added the artificial hyper-viscosity (−∆) ℓ/2 , ℓ > 2, to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t → ∞ toward solutions of the original Navier-Stokes system.

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Section: Theorem 24 Denote By U(x T) and W(x T) The Solutions To Tmentioning

confidence: 99%

About the Regularized Navier?Stokes Equations

Cannone

Karch

2005

J. math. fluid mech.

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“…Then, the fact w n is identically zero near r = 1 implies that ψ n also vanishes near r = 1 due to the equation (32). With the notion of f = f n = f r,n e inθ e r + f θ,n e inθ e θ , we see…”

Section: Biot-savart Law Inmentioning

confidence: 96%

“…The existence theory in the Lorenz spaces has been established by Kozono and Yamazaki [33]. The local L r stability of these stationary solutions is established in [3] and Kozono and Yamazaki [32], while the global L 2 stability is shown in [3]. The reader is referred to recent results by Karch, Pilarczyk, and Schonbek [28] and Hishida and Schonbek [24], where the global L 2 stability of small global solutions in L ∞ (0, ∞; L 3,∞ σ (R 3 )) is obtained.…”

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

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On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Maekawa

2017

Arch Rational Mech Anal

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We study the stability of some exact stationary solutions to the two-dimensional Navier-Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations.Keywords Navier-Stokes equations · two-dimensional exterior flows · scale-criticality · stability of stationary flows · flow around a rotating obstacle

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“…From the stability point of view, several papers are devoted to study of the stability of the stationary solutions for the particular case of the Navier-Stokes equations, see for instance, Kozono & Ozawa [15], Borchers & Miyakawa [1], Chen [2], Kozono & Ogawa [14], Kozono & Yamazaki [16]. In general, these studies have been made using a characterization of the fractional power of the associated linearized operator and estimates of kind L p − L r for the semigroup.…”

Section: Introductionmentioning

confidence: 99%

Global existence and exponential stability for the micropolar fluid system

Villamizar-Roa

Rodríguez-Bellido

2007

Z. angew. Math. Phys.

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We consider the micropolar fluid system in a bounded domain of R 3 and prove the existence and the uniqueness of a global strong solution with initial data being a perturbation of the stationary solution, whose existence is also obtained. We prove that these solutions converge uniformly to the stationary solutions with exponential decay rate. The technique of our analysis is the semigroups approach in L p −spaces.

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On a larger class of stable solutions to the Navier-Stokes equations in exterior domains

Cited by 61 publications

References 31 publications

About the Regularized Navier?Stokes Equations

About the Regularized Navier?Stokes Equations

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Global existence and exponential stability for the micropolar fluid system

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