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 more general boundary condition which includes Taylor-Couette type flows. Several applications are presented as well. ∗wang@math.iastate.edu, 1-(515)294-1752 (T), 1-(515)294-5454(F)

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