A note on Stokes approximations to Leray solutions of the incompressible Navier–Stokes equations in Rn

Abstract

In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t − (n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·, t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·, t) will always decay at least as fast as t − (n+2)/4, no matter how slow the decay of ku(·, t) kL 2 (Rn ) might be.