  A basic example of shear flow was introduced by DiPerna and Majda to study
the weak limit of oscillatory solutions of the Euler equations of
incompressible ideal fluids. In particular, they proved by means of this
example that weak limit of solutions of Euler equations may, in some cases,
fail to be a solution of Euler equations. We use this shear flow example to
provide non-generic, yet nontrivial, examples concerning the loss of smoothness
of solutions of the three-dimensional Euler equations, for initial data that do
not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow
example the existence of weak solutions for the three-dimensional Euler
equations with vorticity that is having a nontrivial density concentrated on
non-smooth surface. This is very different from what has been proven for the
two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the
real analyticity of the interface. Eventually, we use this shear flow to
provide explicit examples of non-regular solutions of the three-dimensional
Euler equations that conserve the energy, an issue which is related to the
Onsager conjecture.
