A basic example of shear flow wasintroduced by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.
This is a joint work with Claude Bardos.