Global analyticity up to the boundary of solutions of the navier‐stokes equation

Komatsu, Gen

doi:10.1002/cpa.3160330405

Cited by 33 publications

(14 citation statements)

References 13 publications

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“…His proof is based on the implicit function theory. His method was improved by several researchers, see [5,7,9,13,17]. But, in order to get the spatial-analyticity only, one can relax the assumption of f .…”

Section: Statement Of the Resultsmentioning

confidence: 97%

On analyticity rate estimates of the solutions to the Navier–Stokes equations in Bessel-potential spaces

Sawada

2005

Journal of Mathematical Analysis and Applications

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On this paper spatial analyticity of solutions to the nonstationary incompressive Navier-Stokes flow inḢ n/2−1 (R n ) is established. The proof is based on the estimates for the higher order derivatives of solutions. These estimates imply not only the regularizing rates near t = 0 but also decaying rates at t → ∞, as long as the solution exists. Although basic strategy is similar to our previous work with Giga for L n space, one can make the proof short using several tools from harmonic analysis.  2005 Elsevier Inc. All rights reserved.

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Section: Statement Of the Resultsmentioning

confidence: 97%

On analyticity rate estimates of the solutions to the Navier–Stokes equations in Bessel-potential spaces

Sawada

2005

Journal of Mathematical Analysis and Applications

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“…We are unable to give a complete history of this study here, but special mention should be made of the important contributions of Hopf ( 1951), Kiselev and Ladyzhenskaya (1957), Serrin (1962), Fujita and Kato (1964), Masuda (1967), Komatsu (1980), and Caffarelli, Kohn, and Nirenberg (1982). Additional references can be found in Giga (1988).…”

mentioning

confidence: 99%

Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions

Raugel¹,

Sell²

1993

J. Amer. Math. Soc.

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We examine the Navier-Stokes equations (NS) on a thin 3 3 -dimensional domain Ω ε = Q 2 × ( 0 , ε ) {\Omega _\varepsilon } = {Q_2} \times (0,\varepsilon ) , where Q 2 {Q_2} is a suitable bounded domain in R 2 {\mathbb {R}^2} and ε \varepsilon is a small, positive, real parameter. We consider these equations with various homogeneous boundary conditions, especially spatially periodic boundary conditions. We show that there are large sets R ( ε ) \mathcal {R}(\varepsilon ) in H 1 ( Ω ε ) {H^1}({\Omega _\varepsilon }) and S ( ε ) \mathcal {S}(\varepsilon ) in W 1 , ∞ ( ( 0 , ∞ ) , L 2 ( Ω ε ) ) {W^{1,\infty }}((0,\infty ),{L^2}({\Omega _\varepsilon })) such that if U 0 ∈ R ( ε ) {U_0} \in \mathcal {R}(\varepsilon ) and F ∈ S ( ε ) F \in \mathcal {S}(\varepsilon ) , then (NS) has a strong solution U ( t ) U(t) that remains in H 1 ( Ω ε ) {H^1}({\Omega _\varepsilon }) for all t ≥ 0 t \geq 0 and in H 2 ( Ω ε ) {H^2}({\Omega _\varepsilon }) for all t > 0 t > 0 . We show that the set of strong solutions of (NS) has a local attractor A ε {\mathfrak {A}_\varepsilon } in H 1 ( Ω ε ) {H^1}({\Omega _\varepsilon }) , which is compact in H 2 ( Ω ε ) {H^2}({\Omega _\varepsilon }) . Furthermore, this local attractor A ε {\mathfrak {A}_\varepsilon } turns out to be the global attractor for all the weak solutions (in the sense of Leray) of (NS). We also show that, under reasonable assumptions, A ε {\mathfrak {A}_\varepsilon } is upper semicontinuous at ε = 0 \varepsilon = 0 .

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“…[8] for the Navier-Stokes equations and [16,21,23] for other models). On domains with boundary, analyticity up to the boundary was obtained in the fundamental works [13,14] (see also [9,15]). These classical works achieve real-analyticity based on an an induction scheme on the number of derivatives.…”

Section: Introductionmentioning

confidence: 95%

A direct approach to Gevrey regularity on the half-space

2018

Partial Differential Equations in Fluid Mechanics

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Global analyticity up to the boundary of solutions of the navier‐stokes equation

Abstract: IntroductionThe present note is concerned with the analyticity of solutions (u, p ) of the nonstationary Navier-Stokes equation

Cited by 33 publications

References 13 publications

On analyticity rate estimates of the solutions to the Navier–Stokes equations in Bessel-potential spaces

On analyticity rate estimates of the solutions to the Navier–Stokes equations in Bessel-potential spaces

Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions

A direct approach to Gevrey regularity on the half-space

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