Numerical verification of regularity in the three-dimensional Navier-Stokes equations

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Current theoretical results for the three-dimensional Navier--Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy (Du₀²:=∫| curl u₀|²) is sufficiently small, Du₀²≤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, Du₀²≤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.