Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
September 21-25, 2009

CSIC Building (#406), Seminar Room 4122.

Examples for Loss of smoothness and energy conservation in the 3D Euler equations

Dr. Claude Bardos

University of Paris VII

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 C1,α} showing that the space C1 is critical for the Euler equations. The role of the space C1 as the critical case is underlined by an example of Pak and Park showing that the problem is well posed in the Besov space B1∞,1 (one has C1,α ⊂ B1∞,1 ⊂ C1 and by the extension of our instability results to the Besov space B1∞,∞ which satisfies the inclusion C1 ⊂ B1∞,∞ ⊂ Cα 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)