Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

Giovanni P. Galdi

doi:10.3934/dcdss.2013.6.1237

Article Contents

2013, Volume 6, Issue 5: 1237-1257. Doi: 10.3934/dcdss.2013.6.1237

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Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

Giovanni P. Galdi^1,

1.
Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261

Received: November 2011

Revised: February 2012

Published: October 2013

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Abstract

We consider the two-dimensional motion of a Navier-Stokes liquid in the whole plane, under the action of a time-periodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding time-periodic flow of period $T$ in an appropriate function class.

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Mathematics Subject Classification: Primary: 35Q30, 76D; Secondary: 76M.

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