Remark on the L<sub>q</sub>− L<sub>∞</sub>estimate of the Stokes semigroup in a 2-dimensional exterior domain

We show that the L p spatial-temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier-Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443-458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, submitted for publication].For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier-Stokes solutions for general L p space when the initial velocity is in L 2 σ ∩ L r , 1 < r q < ∞ (1 < r < q = ∞).

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“…The temporal decay rates for the 2D Navier-Stokes is known by Kozono and Ogawa [22] and Dan and Shibata [14]:…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Bae

Jin²

2006

Journal of Functional Analysis

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“…Let a ∈ L 2 σ ( ), and let u be the solution of (1.1). The decay rates for the 2D Navier-Stokes flows u are shown by Kozono and Ogawa [30], Dan and Shibata [16], respectively:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

Han

2011

manuscripta math.

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“…Schonbek [24,25] also considered the decay rates on the whole space R n under some restrictions on u 0 . Dan and Shibata [10,11] established the L q -L r estimates for the Stokes flow of (1.1) on the exterior domains. Giga, Matsui and Shimizu [16] showed the decay rate for the first derivatives of the Stokes flow u of (1.1) in L 1 (R n + ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces

Han¹

2010

Journal of Differential Equations

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Using the solution formula in Ukai (1987) [27] for the Stokes equations, we find asymptotic profiles of solutions in L 1 (R n + ) (n 2) for the Stokes flow and non-stationary Navier-Stokes equations.Since the projection operator P : L 1 (R n + ) → L 1 σ (R n + ) is unbounded, we use a decomposition for P (u · ∇u) to overcome the difficulty, and prove that the decay rate for the first derivatives of the strong solution u of the Navier-Stokes system in L 1 (R n + ) is controlled by2 ) for any t > 0.

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Remark on the L_q− L_∞estimate of the Stokes semigroup in a 2-dimensional exterior domain

Abstract: We proved L q − L ∞ type estimates of the Stokes semigroup in a 2-dimensional exterior domain. Our proof is based on the investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

Cited by 45 publications

References 14 publications

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces

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Remark on the Lq− L∞estimate of the Stokes semigroup in a 2-dimensional exterior domain

Abstract: We proved L q − L ∞ type estimates of the Stokes semigroup in a 2-dimensional exterior domain. Our proof is based on the investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

Cited by 45 publications

References 14 publications

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

Asymptotic behavior for the Stokes flow and Navier–Stokes equations in half spaces

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Remark on the L_q− L_∞estimate of the Stokes semigroup in a 2-dimensional exterior domain