LOSS OF SMOOTHNESS AND ENERGY CONSERVING ROUGH WEAK SOLUTIONS FOR THE 3d EULER EQUATIONS

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 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 C1,α. In addition, 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. Moreover, we show by means of this shear flow example that, unlike to the two-dimensional case, the minimal regularity in the three-dimensional vortex sheet Kelvin-Helmholtz problem need not to be the class of real analytic solutions. This paper is dedicated to Professor V. Solonnikov, on the occasion of his 75th birthday, as token of friendship and admiration for his contributions to research in partial differential equations and fluid mechanics. MSC Classification: 76 F02, 76 B03.