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

Abstract

Current theoretical results for the three-dimensional Navier--Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy ($\Vert D{u}_0{\Vert }^2:=\int |\mathrm{curl}{u}_0{|}^2$) is sufficiently small, $\Vert D{u}_0{\Vert }^2\le {\chi }_0$. 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, $\Vert D{u}_0{\Vert }^2\le {\chi }_1$. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

References in corpus (1)

• An a posteriori condition on the numerical approximations of the Navier-Stokes equations for the existence of a strong solution