On a $\nu$-continuous family of strong solutions to the Euler      or Navier-Stokes equations with the Navier-Type boundary      condition

Hamid Bellout; Jiří Neustupa; Patrick Penel

doi:10.3934/dcds.2010.27.1353

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2010, Volume 27, Issue 4: 1353-1373. Doi: 10.3934/dcds.2010.27.1353

This issue Previous Article Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations Next Article Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness

On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition

1.
Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115
2.
Czech Academy of Sciences, Mathematical Institute, Žitná 25, 115 67 Prague 1
3.
Université du Sud Toulon–Var, Dép. Mathématique et Laboratoire SNC, BP 20132, 83957 La Garde Cedex

Received: June 2009

Revised: February 2010

Published: December 2010

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Abstract

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T₀ > 0, V^* > 0 and a unique family of strong solutions u^v of the Euler or Navier-Stokes initial-boundary value problem on the time interval (0, T₀), depending continuously on the viscosity coefficient $\nu$ for $0\leq\nu< $ V^*. The solutions of the Navier-Stokes problem satisfy a Navier-type boundary condition. We give the information on the rate of convergence of the solutions of the Navier-Stokes problem to the solution of the Euler problem for $\nu\to 0+$.

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Mathematics Subject Classification: 35Q30, 35Q31, 76D05, 76D09.

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On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition

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