A Missed Persistence Property for the Euler Equations, and its Effect on Inviscid Limits

摘要

We consider the problem of the strong convergence, as the viscosity goes tozero, of the solutions to the three-dimensional evolutionary Navier-Stokesequations under a Navier slip-type boundary condition to the solution of theEuler equations under the zero flux boundary condition. In spite of thearbitrarily strong convergence results proved in the flat boundary case, see[4], it was shown in reference [5] that the result is false in general, byconstructing an explicit family of smooth initial data in the sphere, for whichthe result fails. Our aim here is to present a more general, simpler andincisive proof. In particular, counterexamples can be displayed in arbitrary,smooth, domains. As in [5], the proof is reduced to the lack of a suitablepersistence property for the Euler equations. This negative result is proved bya completely different approach.