We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite
-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made by analytical non-blowup criteria introduced by Deng et. al [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase to rule out singular behavior.
We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 0
26306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.
