Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in R-n

We present a new derivation of upper bounds for the decay of higher order d erivatives of solutions to the unforced Navier-Stokes equations in R-n. The method, based on so-called Gevrey estimates, also yields explicit bounds o n the growth of the radius of analyticity of the solution in time. Moreover , under the assumption that the Navier-Stokes solution stays sufficiently c lose to a solution of the heat equation in the L-2 norm-a result known to b e true for a large class of initial data-lower bounds on the decay of highe r order derivatives can be obtained. (C) 2000 Academic Press.