A discrete Kato type theorem on inviscid limit of Navier-Stokes flows
Wenfang (Wendy) Cheng, Xiaoming Wang
The inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a result indicating the difficulty in numerical study of the problem. More precisely, we show that numerical solutions of the incompressible Navier-Stokes equations converge to the exact solution of the Euler equations at vanishing viscosity provided that small scales of the order of $\nu/U$ in the directions tangential to the boundary in an appropriate boundary layer is not resolved in the scheme. Here $\nu$ is the kinematic viscosity of the fluid and $U$ is the typical velocity taken to be the maximum of the shear velocity at the boundary for the inviscid flow. Such a result is somewhat surprising since such a small scale is much smaller than any of the known small scales predicted by conventional theory of turbulence and boundary layer theory. On the other hand, such a result can be viewed as a discrete version of our previous result (Wang 2001) which generalized earlier result of Kato (1984) where the relevance of a scale proportional to the kinematic viscosity to the problem of vanishing viscosity was first discovered.