Analytic Study of Shell Models of Turbulence
  In this paper we study analytically the viscous `sabra' shell model of 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 exponentially decaying energy dissipation range for the sufficiently smooth forcing. 
Peter Constantin, Boris Levant and Edriss S. Titi