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 ω and another material vector W such that γ = ω ×W,
where γ 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 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 GaussWeingarten
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.
An addendum for this article has been published in 2018 Nonlinearity 31, 3973.
http://eprints.whiterose.ac.uk/134288/
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