A Kato type theorem on zero viscosity limit of Navier-Stokes flows

great admiration and friendship on the occasion of the retirement of Professor Foias and on the occasion of the sixtieth birthday of Professor Temam 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 conver-gence 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 ve-locity 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 re-quire 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.