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

Pages: 1234 - 1243, Issue Special, September 2011

 Abstract        Full Text (334.1K)              

Joanna Rencławowicz - Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland (email)
Wojciech M. Zajączkowski - Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland (email)

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.

Keywords:  Navier-Stokes equation, weighted Sobolev spaces, Neumann boundary- value problem, Dirichlet boundary-value problem, global solutions, large flux
Mathematics Subject Classification:  Primary: 35Q30; Secondary: 76D03, 76D05

Received: July 2010;      Revised: March 2011;      Published: October 2011.