Global regular solutions to the Navier-Stokes equations with large flux

Joanna Rencławowicz; Wojciech M. Zajączkowski

doi:10.3934/proc.2011.2011.1234

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2011, Volume 2011: 1234-1243. Doi: 10.3934/proc.2011.2011.1234 open access

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Global regular solutions to the Navier-Stokes equations with large flux

Joanna Rencławowicz^1, and
Wojciech M. Zajączkowski^2,

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw
2.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw

Received: June 30, 2010

Revised: February 28, 2011

Published: August 2017

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Abstract

We consider the Navier-Stokes motion in a bounded cylinder with boundary slip conditions. We assume an inflow and an outflow of the fluid through the bottom and the top of the cylinder where the magnitude of the flux is not restricted. We require that the derivatives of the initial velocity and the external force with respect to the variable along the axis of the cylinder are sufficiently small. Under these conditions we are able to prove global existence of regular solutions. Since we are interested in nonvanishing in time flux we need to use the Hopf function to derive global energy estimate.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D05.

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