On the global stability of smooth solutions of the Navier-Stokes equations

In Morosi and Pizzocchero (2015) and previous papers by the same authors, a general smooth setting was proposed for the incompressible Navier-Stokes (NS) Cauchy problem on a torus of any dimension d greater than or equal to 2, and the a posteriori analysis of its approximate solutions. In this note, using the same setting I propose an elementary proof of the following statement: global existence and time decay of the NS solutions are stable properties with respect to perturbations of the initial datum. Fully explicit estimates are derived, using Sobolev norms of arbitrarily high order. An application is proposed, in which the initial data are generalized Beltrami flows. A comparison with the related literature is performed.