Numerical verification of regularity in the three-dimensional Navier–Stokes equations for bounded sets of initial data

Article type: Research Article

Authors: Robinson, James C. | Sadowski, Witold

Affiliations: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK | Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, and Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Abstract: Current theoretical results for the three-dimensional Navier–Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy (‖Du0‖2:=∫|curl u0|2) is sufficiently small, ‖Du0‖2≤χ0. In fact, this smallness condition is such that the enstrophy is always non-increasing. In this paper we provide a numerical procedure that will verify regularity of solutions for any bounded set of initial conditions, ‖Du0‖2≤χ1. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

Keywords: regularity of the Navier–Stokes equations

DOI: 10.3233/ASY-2008-0899

Journal: Asymptotic Analysis, vol. 59, no. 1-2, pp. 39-50, 2008