A Kato Type Theorem on Zero Viscosity Limit of Navier-Stokes Flows
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.
X. Wang, "A Kato Type Theorem on Zero Viscosity Limit of Navier-Stokes Flows," Indiana University Mathematics Journal, vol. 50, no. SUPPL., pp. 223 - 241, Indiana University Mathematics Journal, Jan 2001.
The definitive version is available at https://doi.org/10.1512/iumj.2001.50.2098
Mathematics and Statistics
Keywords and Phrases
Boundary Layer; Energy Dissipation Rate; Euler Equations; Navier-Strokes Equations; Zero Viscosity Limit
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2001