Abstract: In statistical theory of turbulence phenomena like, instability with respect to the initial data, roughness of the solution, and dissipation of energy are very closely related. Therefore we use the shear flow to show that the situation is radically different for individual solutions of the incompressible Euler equations. The 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 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,α}} showing that the space C^{1} is critical for the Euler equations. The role of the space C^{1} as the critical case is underlined by an example of Pak and Park showing that the problem is well posed in the Besov space B^{1}_{∞,1} (one has C^{1,α} ⊂ B^{1}_{∞,1} ⊂ C^{1} and by the extension of our instability results to the Besov space B^{1}_{∞,∞} which satisfies the inclusion C^{1} ⊂ B^{1}_{∞,∞} ⊂ C^{α} Moreover, we show the existence of solutions with vorticity having a non trivial density on non smooth surface. 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 related to the Onsager conjecture. It may be interesting to compare the properties of the family of shear flow solutions with the "wild solutions" constructed (not explicitely) by C. De Lellis and L. Szekelyhidi. There one has an infinite family of solutions for the same initial data. They are "wild" in the sense that they are limit on oscillating velocity fields and they conserve the energy. In relation view an "optimistic" construction of a measure for statistical theory of turbulence one could imagine that in both cases one has family of solutions of measure zero. (In spite of the fact that in the second case the wild solutions form a residual set) |