Inertial Manifolds for Certain Sub-Grid Scale $\alpha$-Models of   Turbulence
  In this note we prove the existence of an inertial manifold, i.e., a global invariant, exponentially attracting, finite-dimensional smooth manifold, for two different sub-grid scale $\alpha$-models of turbulence: the simplified Bardina model and the modified Leray-$\alpha$ model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of the large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differential equations. 
Mohammad Abu Hamed, Yanqiu Guo, and Edriss S. Titi
math.DS