Study of the 3D Euler equations using Clebsch potentials: dual mechanisms for geometric depletion

After surveying analyses of the 3D Euler equations using the Clebsch potentials scattered over the literature, we report some preliminary new results.

1. Assuming that flow fields are free from nulls of the impulse and the vorticity fields, we study how constraints imposed by the Clebsch potentials lead to a degenerate geometrical structure, typically in the form of depletion of nonlinearity. We consider a vorticity surface spanned by $\boldsymbol {\omega}$ and another material vector $\boldsymbol {W}$ such that $\boldsymbol {\gamma}=\boldsymbol {\omega}\times \boldsymbol {W},$ where $\boldsymbol {\gamma}$ is the impulse variable in geometric gauge. We identify dual mechanism for geometric depletion and show that at least of one them is acting if $\boldsymbol {W}$ does not develop a null. This suggests that formation of singularity in flows endowed with Clebsch potentials is less likely to happen than in more general flows. Some arguments are given towards exclusion of 'type I' blowup. A mathematical challenge remains to rule out singularity formation for flows which have Clebsch potentials everywhere.

2. We exploit classical differential geometry kinematically to write down the Gauss–Weingarten equations for the vorticity surface of the Clebsch potential in terms of fluid dynamical variables, as are the first, second and third fundamental forms. In particular, we derive a constraint on the size of the Gaussian curvature near the point of a possible singularity. On the other hand, an application of the Gauss–Bonnet theorem reveals that the tangential curvature of the surface becomes large in the neighborhood of near-singularity.

3. Using spatially-periodic flows with highly-symmetry, i.e. initial conditions of the Taylor–Green vortex and the Kida–Pelz flow, we present explicit formulas of the Clebsch potentials with exceptional singular surfaces where the Clebsch potentials are undefined. This is done by connecting the known expressions with the solenoidal impulse variable (i.e. the incompressible velocity) using suitable canonical transforms. By a simple argument we show that they keep forming material separatrices under the time evolution of the 3D Euler equations. We argue on this basis that a singularity, if developed, will be associated with these exceptional material surfaces. The difficulty of having Clebsch potentials globally on all of space have been with us for a long time. The proposal rather seeks to turn the difficulty into an advantage by using their absence to identify and locate possible singularities.