Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier–Stokes Equations in Rn

We present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced Navier–Stokes equations in Rn. The method, based on so-called Gevrey estimates, also yields explicit bounds on the growth of the radius of analyticity of the solution in time. Moreover, under the assumption that the Navier–Stokes solution stays sufficiently close to a solution of the heat equation in the L2 norm—a result known to be true for a large class of initial data—lower bounds on the decay of higher order derivatives can be obtained.