A Kato Type Theorem on Zero Viscosity Limit of Navier-Stokes Flows

Abstract

We present a necessary and sufficient condition for the convergence of solutions of the incompressible Navier-Stokes equations to that of the Euler equations at vanishing viscosity. Roughly speaking, convergence is true in the energy space if and only if the energy dissipation rate of the viscous flows due to the tangential derivatives of the velocity in a thick enough boundary layer, a small quantity in classical boundary layer theory, approaches zero at vanishing viscosity. This improves a previous result of T. Kato (1984), in the sense that we require tangential derivatives only, while the total gradient is needed in Kato's work. However, we require a slightly thicker boundary layer. We also improve our previous result, where only sufficient conditions were obtained. Moreover, we treat a more general boundary condition, which includes Taylor-Couette type flows. Several applications are presented as well.

Department(s)

Mathematics and Statistics

Keywords and Phrases

Boundary Layer; Energy Dissipation Rate; Euler Equations; Navier-Strokes Equations; Zero Viscosity Limit

International Standard Serial Number (ISSN)

0022-2518

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2023 Indiana University Mathematics Journal, All rights reserved.

Publication Date

01 Jan 2001

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