Singularity Formation in 3-D Vortex Sheets

The celebrated model of Fermi, Pasta and Ulam is revisited with the aim of investigating the relaxation of the system towards equipartition. We perform an extensive numerical exploration starting with all the energy E initially concentrated on the lowest frequency mode, as in the original FPU report. We produce evidence of the existence of two well separated different time scales. In a short time we observe the formation of a packet of low frequency modes which share most of the total energy,while the high frequency modes appear to be frozen in a exponentially decreasing distribution. After this fast relaxation a second phase begins during which the energy flows at a definitely slower rate towards the higher frequency modes. The numerical exploration shows that the fraction of the total number N of modes that take active part in the formation of the initial packet is a function of the specific energy E/N. Furthermore, the speed of the energy flow during the second phase tends quite rapidly to zero with E/N. Thus, this phenomenon appears to be relevant for the thermodynamic limit