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November  2010, 27(4): 1353-1373. doi: 10.3934/dcds.2010.27.1353

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

Hamid Bellout 1, , Jiří Neustupa 2, and  Patrick Penel 3,

1. 

Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115, United States
2. 

Czech Academy of Sciences, Mathematical Institute, Žitná 25, 115 67 Prague 1, Czech Republic
3. 

Université du Sud Toulon–Var, Dép. Mathématique et Laboratoire SNC, BP 20132, 83957 La Garde Cedex, France

Received  June 2009 Revised  February 2010 Published  March 2010

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 uv 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 $\nu\to 0+$.
Keywords: Navier-Stokes equations, zero viscosity limit., Euler equations.
Mathematics Subject Classification: 35Q30, 35Q31, 76D05, 76D0.
Citation: Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
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