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Xinyu Zhao: Systematic search for singularities in 3D Euler flows

Time: Thu 2023-02-23 14.00 - 15.00

Location: Zoom

Video link: Meeting Id: 632 9929 3349

Participating: Xinyu Zhao (McMaster U)

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The local well-posedness of smooth solutions of 3D incompressible Euler equations has been established when the initial data is in the Sobolev space $H^s$ for $s>5/2$. However, it is still an open question whether these solutions develop finite-time singularities. Today, we will present a numerical study of this question where we systematically search for initial data that may lead to potential singularity through PDE-constrained optimization. The optimization problem is solved numerically using a state-of-the-art Riemannian conjugate gradient method where the Sobolev gradients are obtained through an adjoint method. The behavior of the obtained extreme flow, which features two colliding distorted vortex rings, suggests a finite-time singularity formation. This is based on a joint work with Bartosz Protas.