Asymptotic analysis of the Stokes problem on general bounded domains: the case of a characteristic boundary

Abstract

The goal of this article is to study the asymptotic behaviour of the solutions of linearized Navier–Stokes equations (LNSE), when the viscosity is small, in a general (curved) bounded and smooth domain in ℝ³ with a characteristic boundary. To handle the difficulties due to the curvature of the boundary, we first introduce a curvilinear coordinate system which is adapted to the boundary. Then we prove the existence of a strong corrector for the LNSE. More precisely, we show that the solution of LNSE behaves like the corresponding Euler solution except in a thin region, near the boundary, where a certain heat solution is added as a corrector.

Keywords:

• boundary layers
• singular perturbations
• Stokes equations
• curvilinear coordinates

AMS Subject Classifications::

• 35B25
• 35C20
• 35K05
• 76D07
• 76D10

Acknowledgements

This work was supported in part by NSF grants DMS 0604235 and 0906440, and by the Research Fund of Indiana University.