John D. Gibbon
Afilliation: Department of Mathematics, Imperial College, London

Title: Conditional Regularity of Solutions of the Three Dimensional Navier-Stokes Equations and Implications for Intermittency

Abstract: An unusual conditional regularity proof is presented for the three-dimensional forced Navier-Stokes equations from which a realistic picture of intermittency emerges. Based on $L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$ for $m\geq 1$, the time integrals $\int_{0}^{t}\Omega_{m}^{\alpha_{m}}d\tau$ with $\alpha_{m} = 2m/(4m-3)$, play a key role in bounding the dissipation from below. By imposing a lower bound on $\int_{0}^{t}\Omega_{m+1}^{\alpha_{m+1}}d\tau$ it is shown that $\Omega_{m}(t)$ cannot become singular for large initial data. By considering movement in the value of $\int_{0}^{t}\Omega_{m+1}^{\alpha_{m+1}}d\tau$ across this imposed critical lower bound, it is shown how solutions behave intermittently, in analogy with a relaxation oscillator. A cascade assumption is also considered.

Slides: GibbonJohn.pdf