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July, 2015 Time periodic solutions of the Navier–Stokes equations with the time periodic Poiseuille flow under (GOC) for a symmetric perturbed channel in $\mathbb{R}^2$
Teppei KOBAYASHI
J. Math. Soc. Japan 67(3): 1023-1042 (July, 2015). DOI: 10.2969/jmsj/06731023

Abstract

Beirão da Veiga [ 5] proves that for a straight channel in $\mathbb{R}^n$ ($n\ge2$) and for a given time periodic flux there exists a unique time periodic Poiseuille flow. As a by product, existence of the time periodic Poiseuille flow in perturbed channels (Leray's problem) is shown for the Stokes problem ($n\ge2$) and for the Navier–Stokes problem ($n\le4$). Concerning the Navier–Stokes case, in [ 5] a quantitative condition required to show the existence of the solutions depends not just on the flux of the time periodic Poiseuille flow but also on the domain itself.

Kobayashi [ 16], [ 18] proves that for a perturbed channel in $\mathbb{R}^n$ $(n=2,3)$ there exists a time periodic solution of the Navier–Stokes equations with the Poiseuille flow applying the theory of the steady problem to the time periodic problem.

In this paper, applying Fujita [ 8] and Kobayashi [ 18], we succeed in proving the existence of a time periodic solution for a symmetric perturbed channel in $\mathbb{R}^2$.

Citation

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Teppei KOBAYASHI. "Time periodic solutions of the Navier–Stokes equations with the time periodic Poiseuille flow under (GOC) for a symmetric perturbed channel in $\mathbb{R}^2$." J. Math. Soc. Japan 67 (3) 1023 - 1042, July, 2015. https://doi.org/10.2969/jmsj/06731023

Information

Published: July, 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1330.35290
MathSciNet: MR3376576
Digital Object Identifier: 10.2969/jmsj/06731023

Subjects:
Primary: 35Q30, 76D05

Rights: Copyright © 2015 Mathematical Society of Japan

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Vol.67 • No. 3 • July, 2015
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