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

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

2007-01-09

arxiv.org/abs/math/0701268v1

Asymptotic Analysis 59 (2008) 39-50

Mathematics

Analysis of PDEs

Scientific paper

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, $\|Du_0\|^2\le\chi_0$. 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, $\|Du_0\|^2\le\chi_1$. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.

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