Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

Abstract

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on $L^{2m}$-norms of the vorticity, denoted by ${\Omega }_m(t)$, and particularly on $D_m={\Omega }_m^{{\alpha }_m}$, where ${\alpha }_m=2m/(4m-3)$ for $m\ge 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1\le m<\infty$ for which the integral condition is satisfied ($Z_m=D_{m+1}/D_m$) $${\int }_0^t\ln (\frac{1+Z_m}{c_{4,m}})d\tau \ge 0$$ then no singularity can occur on $[0,t]$. The constant $c_{4,m}\searrow 2$ for large $m$. Secondly, for the forced case, by imposing a critical \textit{lower} bound on ${\int }_0^t{D}_{m}d\tau$, no singularity can occur in $D_m(t)$ for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive ${\int }_0^t{D}_{m}d\tau$ over this critical value can be ruled out whereas other types cannot.

A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figure

References in corpus (1)

• The multifractal nature of turbulent energy dissipation

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