Systematic search for singularities in 3D Euler flows

Xinyu Zhao, McMaster University

01/30, 2022 at 11:30am-12:30pm (HH312)

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.