Journal of Mathematical Fluid Mechanics

, Volume 13, Issue 1, pp 117–135

Concerning the Wk,p-Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

Article

DOI: 10.1007/s00021-009-0012-3

Cite this article as:
Beirão da Veiga, H. & Crispo, F. J. Math. Fluid Mech. (2011) 13: 117. doi:10.1007/s00021-009-0012-3

Abstract

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces Wkp(Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beirão da Veiga and Crispo in J Math Fluid Mech, 2009, doi:10.1007/s00021-009-0295-4). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in Lp-spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.

Mathematics Subject Classification (2000)

35Q30 76D05 76D09 

Keywords

Stokes and Navier–Stokes equations inviscid limit slip boundary conditions 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics “U. Dini”University of PisaPisaItaly

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