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Systematic search for singularities in 3D Euler flows

Xinyu ZhaoBartosz Protas
Dec 2022
We consider the question whether starting from a smooth initial condition 3Dinviscid Euler flows on a periodic domain $\mathbb{T}^3$ may developsingularities in a finite time. Our point of departure is the well-known resultby Kato (1972), which asserts the local existence of classical solutions to theEuler system in the Sobolev space $H^m(\mathbb{T}^3)$ for $m > 5/2$. Thus,potential formation of a singularity must be accompanied by an unbounded growthof the $H^m$ norm of the velocity field as the singularity time is approached.We perform a systematic search for "extreme" Euler flows that may realize sucha scenario by formulating and solving a PDE-constrained optimization problemwhere the $H^3$ norm of the solution at a certain fixed time $T > 0$ ismaximized with respect to the initial data subject to suitable normalizationconstraints. This problem is solved using a state-of-the-art Riemannianconjugate gradient method where the gradient is obtained from solutions of anadjoint system. Computations performed with increasing numerical resolutionsdemonstrate that, as asserted by the theorem of Kato (1972), when theoptimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remainsbounded in the extreme flows found by solving the optimization problem, whichindicates that the Euler system is well-posed on this "short" time interval. Onthe other hand, when the window $[0, T]$ is long, possibly longer than the timeof the local existence asserted by Kato's theorem, then the $H^3$ norm of theextreme flows diverges upon resolution refinement, which indicates a possiblesingularity formulation on this "long" time interval. The extreme flow obtainedon the long time window has the form of two colliding vortex rings and ischaracterized by certain symmetries. In particular, the region of the flow inwhich a singularity might occur is nearly axisymmetric.