An invisciddyadic model of turbulence: The global attractor

Abstract

Properties of an infinite system of nonlinearly coupledordinary differential equations are discussed. Thissystem models some properties present in the equationsof motion for an inviscid fluid such as the skew symmetryand the 3-dimensional scaling of the quadratic nonlinearity.In a companion paper [8] it is proved thatevery solution for the system with forcingblows up in finite time in the Sobolev $H^{5/6}$ norm.In this present paper, it is proved that after theblow-up time all solutions stay in $H^s$, $s < 5/6$for almost all time. It is proved that the model systemexhibits the phenomenon of anomalous (or turbulent) dissipationwhich was conjectured for the Euler equations by Onsager.As a consequence of this anomalous dissipation the unique equilibriumof the system is a global attractor.