Systematic search for singularities in 3D Euler flows on a periodic domain

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 may develop finite-time singularities. In order to systematically search for initial data that may lead to a singularity formation in finite time, we formulate a PDE-constrained optimization problem in which the quantity $|| \boldsymbol{u}(T) ||_{H^3}$, where $\boldsymbol{u}(t)$, $0 \le t \le T$, is the solution of the Euler equation, is maximized subject to the constraint $|| \boldsymbol{u}(0) ||_{H^3} = 1$. This optimization problem is solved numerically using a state-of-the-art Riemannian conjugate gradient method based on Sobolev gradients obtained by solving a suitable adjoint system. In the process, we repeatedly refine the resolution to detect a possible finite-time blow-up indicated by an unbounded growth of the maximized quantity. The behavior of the extreme flows obtained in this way is consistent with the formation of singularities in finite time when $T$ is sufficiently large.