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

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.