Solutions of the 3D Navier–Stokes equations for initial data in : Robustness of regularity and numerical verification of regularity for bounded sets of initial data in

We consider the three-dimensional Navier–Stokes equations on a periodic domain. We give a simple proof of the local existence of solutions in H ̇ 1 / 2 , and show that the existence of a regular solution on a bounded time interval [ 0 , T ] is stable with respect to perturbations of the initial data in H ̇ 1 / 2 and perturbations of the forcing function in L 2 ( 0 , T ; H − 1 / 2 ) . This forms the key ingredient in a proof that the assumption of regularity for all initial conditions in any given ball in H ̇ 1 can be verified computationally in a finite time, strengthening a previous result of Robinson and Sadowski [J.C. Robinson and W. Sadowski, Numerical verification of regularity in the three-dimensional Navier-Stokes equations for bounded sets of initial data, Asymptot. Anal. 59 (2008) 39–50].