Solutions of The 3D Navier-Stokes Equations for Initial Data In H¿1/2: Robustness of Regularity and Numerical Verification of Regularity for Bounded Sets of Initial Data In H¿1

Abstract

We consider the three-dimensional Navier–Stokes equations on a periodic domain. We give a simple proof of the local existence of solutions in View the MathML source, 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 View the MathML source and perturbations of the forcing function in L2(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 View the MathML source 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].