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AIMS Lab Seminar - Di Kang - Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows


HH 403

Speaker: Di Kang

Title: 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 equation --- 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 finite a priori bounds available for 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 $\E_0$ 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 $\E_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to $\E_0^{3/2}$. 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.
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