L2-asymptotic stability of singular solutions to the Navier–Stokes system of equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

Karch, Grzegorz; Pilarczyk, Dominika; Schonbek, María E.

doi:10.1016/j.matpur.2016.10.008

Cited by 24 publications

(43 citation statements)

References 34 publications

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“…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: 84%

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

confidence: 84%

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

Maekawa

2017

Arch Rational Mech Anal

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“…Apply (3.9) with ψ = ζ n − ϕ, with ζ n (x) = n −1 ζ(x/n) is a smooth and compactly supported approximation of the Dirac measure, and let n → ∞ (see [15] [24]).…”

Section: Mathematical Settingsmentioning

confidence: 99%

“…The proof is based on ideas of [15], [24]: We decompose the L 2 -norm of the Fourier transform of the weak solution u as follows , the fundamental solution of the heat equation at t = 1. We estimate separately the low frequencies and the high energy frequencies terms in (3.13).…”

Section: Mathematical Settingsmentioning

confidence: 99%

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Long time behavior for a dissipative shallow water model

Sciacca

Schonbek

Sammartino

2017

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

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Abstract. We consider the two-dimensional shallow water model derived in [20], describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, considering the whole domain R 2 and under suitable conditions on the time dependent forcing term, we prove the L2 asymptotic decay of the weak solutions.

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“…On the topic of asymptotic stability for the solutions, there have been many classical results to the Navier-Stokes equation [1,[4][5][6][7][8]10,[12][13][14]. The energy decay problem of weak solutions to the Navier-Stokes equation was originally suggested by Leray in his pioneering papers [9,10].…”

Section: Introductionmentioning

confidence: 99%