Lagrangian dynamics and regularity of the spin Euler equation

Abstract

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector rather than the velocity. It enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. The spin Euler equation is also equivalent to a special case of the Landau-Lifshitz equation with a specific effective magnetic field, revealing a possible connection between ideal flow and magnetic crystal. We conduct direct numerical simulations of three ideal flows of the vortex knot, vortex link and modified Taylor-Green flow by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale-Kato-Majda criterion in detecting the flows that are incapable of producing finite-time singularities.