  We examine the blow-up claims of the incompressible Euler equations for
several specific flow-fields, (1) the columnar eddies in the vicinity of
stagnation; (2) a quasi-three-dimensional structure for illustrating
oscillations and concentrations in shears; (3) a 2-and-1/2D flow. We assert
that these claimed finite-time singularities are not genuine. Over the whole
space, the potential or the velocity coincides with singularities of harmonic
functions. We show that the existence of the potential for unique velocity is
necessary but insufficient, as the velocity fields are merely specified up to
multiples of the curl of a vector potential, which satisfies a system of
degenerated Laplace's equations. We have derived several closed-form formulas
in simply-connected domains, as well as orthogonal curvilinear co-ordinates. It
follows that steady solutions of Euler's equations are indeterminate. Hence,
use of singular potential flow in expounding fluid motions is inept. The
formulation of the primitive variables (velocity and pressure) gives rise to
non-unique flow fields. In essence, the notion of a finite-time singularity is
an ill-defined proposition. In Appendix A, we show that the well-posedness
claim (Yudovich, 1995) of planar vorticity in Lebesgue's class has been made on
the basis of flawed technicalities. Certain inequalities of functional analysis
play key roles in the regularity study. In Appendix B, we assert that their
derivations have not been rigorously formulated. Concretely, applications of
Sobolev's inequality (as well as Ladyzhenskaya's inequalities) to non-trivial
solenoidal velocity fields are unattainable. There have been claims of singular
flow solutions in the technical literature. However, a large majority of the
deductions rely on these imprecise functional inequalities and hence suffer
from logical inconsistency.
