Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations

Abstract

Higher moments of the vorticity field ${\Omega }_m(t)$ in the form of $L^{2m}$-norms ($1\le m<\infty$) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier-Stokes equations on the domain $[0,L{]}_{per}^3$. It is found that the set of quantities $$D_m(t)={\Omega }_m^{{\alpha }_m},{\alpha }_m=\frac{2m}{4m-3},$$ provide a natural scaling in the problem resulting in a bounded set of time averages $<D_m{>}_T$ on a finite interval of time $[0,T]$. The behaviour of $D_{m+1}/D_m$ is studied on what are called `good' and `bad' intervals of $[0,T]$ which are interspersed with junction points (neutral) ${\tau }_i$. For large but finite values of $m$ with large initial data \big($Ω_{m}(0) \leq \varpi_{0}O(\Gr^{4})$\big), it is found that there is an upper bound $$ Ω_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = νL^{-2}, $$ which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray \cite{Leray} and Scheffer \cite{Scheff76}, this estimate for ${\Omega }_m$ corresponds to a length scale well below the validity of the Navier-Stokes equations.

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References in corpus (3)

• The multifractal nature of turbulent energy dissipation
• Twenty-five years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin
• Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations

Cited by in corpus (4)

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