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

TL;DR

This paper presents new conditional regularity results for 3D Navier-Stokes solutions based on time-integral conditions involving vorticity norms, with implications for understanding intermittency and singularity formation.

Contribution

It introduces novel integral conditions involving vorticity norms that prevent singularities, advancing understanding of solution regularity and intermittency in 3D Navier-Stokes equations.

Findings

Integral condition prevents singularity in unforced case.

Critical lower bound on vorticity norm integral inhibits singularities in forced case.

Solutions exhibit intermittent behavior analogous to relaxation oscillators.

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 $m\geq 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1\leq m < \infty$ for which the integral condition is satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$ \int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0$$ 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.