Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows

Abstract

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy ${\mathcal{E}}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of ${\mathcal{E}}_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to ${\mathcal{E}}_0^{3/2}$ as ${\mathcal{E}}_0$ becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

arXiv comment39 pages, 15 figures; accepted for publication in Journal of Fluid Mechanics

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• Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions
• Systematic search for singularities in 3D Euler flows
• Viscous and Inviscid Reconnection of Vortex Rings on Logarithmic Lattices
• Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System