Inertial Manifolds for the Dyadic Model of the Navier-Stokes Equations

We prove the existence of finite-dimensional inertial manifolds for the dyadic model of turbulence for all dissipation exponents α≥ 1/3 by working in the H^α-norm and employing a generalized cone method. The dimension scales as N~ 1/αlogλlogν⁻¹, matching the optimal upper bounds for shell models. The construction relies on a low-mode cut-off, a forward cascade estimate and a modified strong squeezing property of Koksch [6]. The resulting inertial manifold is Lipschitz and C1+ε-smooth, and satisfies the exponential tracking property. This provides a rigorous finite-dimensional reduction for α≥ 1/3. We also answer an open question by Cheskidov (2008) regarding the existence of strong compact global attractors for α<1/2. We also show global regularity for α<1/2.