Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2011-08-23
Nonlinear Sciences
Chaotic Dynamics
12 pages, 1 table, 1 figure
Scientific paper
Two unusual time-integral conditional regularity proofs 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 $1\leq m < \infty$ and an $0 < \varepsilon < 2$ such that $\int_{0}^{t}D_{m+1}^{\varepsilon}d\tau \geq c_{\varepsilon,m}\int_{0}^{t} D_{m}^{\varepsilon}d\tau$ with $c_{\varepsilon, m} > 1$ then no singularity can occur on $[0,\,t]$. 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 others cannot.
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