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- W2072059094 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 $mgeq 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1leq m < infty$ for which the integral condition is satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$ int_{0}^{t}ln (frac{1 + Z_{m}}{c_{4,m}}) dtau geq 0$$ 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 $int_{0}^{t}D_{m} dtau$, 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 $int_{0}^{t}D_{m} dtau$ over this critical value can be ruled out whereas other types cannot." @default.
- W2072059094 created "2016-06-24" @default.
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- W2072059094 date "2012-09-06" @default.
- W2072059094 modified "2023-09-23" @default.
- W2072059094 title "Conditional regularity of solutions of the three-dimensional Navier-Stokes equations and implications for intermittency" @default.
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- W2072059094 doi "https://doi.org/10.1063/1.4742857" @default.
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