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

J C Robinson; W Sadowski

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arXiv · math/0701268

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

J C Robinson·W Sadowski

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Abstract

Current theoretical results for the three-dimensional Navier--Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy ($\|Du_0\|^2:=\int|{\rm curl} u_0|^2$) is sufficiently small, $\|Du_0\|^2\leχ_0$. 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, $\|Du_0\|^2\leχ_1$. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

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J C Robinson, W Sadowski. 2007-01-09. Numerical verification of regularity in the three-dimensional Navier-Stokes equations. https://arxiv.org/abs/math/0701268

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