Tohoku Mathematical Journal

Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels

Teppei Kobayashi

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Abstract

H. Beirão da Veiga proved that, for a straight channel in $\boldsymbol{R}^n$ ($n$ arbitarily large) and for a given flux with the time periodicity, there exists a unique time periodic Poiseuille flow in a straight channel in $\boldsymbol{R}^n$. Furthermore, the existence of a time periodic solution in a perturbed channel (Leray's problem) is shown for the Stokes problem (arbitary dimension) and for the Navier-Stokes problem ($n\le4$). Concerning the Navier-Stokes case, a quatitative condition requaired to show the existence of a time periodic solution depends not just on the flux of the time periodic Poiseuille flow but also on the domain it self. In this paper, by applying the result of H. Beirão da Veiga and C. J. Amick, we succeed in proving the independence of such a condition on the particular domain.

Article information

Source
Tohoku Math. J. (2), Volume 66, Number 1 (2014), 119-135.

Dates
First available in Project Euclid: 7 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1396875666

Digital Object Identifier
doi:10.2748/tmj/1396875666

Mathematical Reviews number (MathSciNet)
MR3189483

Zentralblatt MATH identifier
1293.35208

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 76D05: Navier-Stokes equations [See also 35Q30]

Keywords
Time periodic solutions of the Navier-Stokes equations the Poiseuille flow channels

Citation

Kobayashi, Teppei. Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels. Tohoku Math. J. (2) 66 (2014), no. 1, 119--135. doi:10.2748/tmj/1396875666. https://projecteuclid.org/euclid.tmj/1396875666


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