Analytic study of shell models of turbulence

In this paper we study analytically the viscous “sabra” shell model of the energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of an exponentially decaying energy dissipation range for sufficiently smooth forcing.