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Two unusual time-integral conditional regularity results are presented forthe three-dimensional Navier-Stokes equations. The ideas are based on$L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$, and particularlyon $D_{m} = \Omega_{m}^{\alpha_{m}}$, where $\alpha_{m} = 2m/(4m-3)$ for $m\geq1$. The first result, more appropriate for the unforced case, can be statedsimply : if there exists an $1\leq m < \infty$ for which the integral conditionis satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$ \int_{0}^{t}\ln (\frac{1 +Z_{m}}{c_{4,m}}) d\tau \geq 0$$ then no singularity can occur on $[0, t]$. Theconstant $c_{4,m} \searrow 2$ for large $m$. Secondly, for the forced case, byimposing a critical \textit{lower} bound on $\int_{0}^{t}D_{m} d\tau$, nosingularity can occur in $D_{m}(t)$ for \textit{large} initial data. Movementacross 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 whereasother types cannot.

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