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 $\alpha\ge 1/3$ by working in the $H^\alpha$-norm and employing a generalized cone method. The dimension scales as $N\sim \frac{1}{\alpha\log\lambda}\log\nu^{-1}$, 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 $C^{1+\epsilon}$-smooth, and satisfies the exponential tracking property. This provides a rigorous finite-dimensional reduction for $\alpha\ge 1/3$. We also answer an open question by Cheskidov (2008) regarding the existence of strong compact global attractors for $\alpha<1/2$. We also show global regularity for $\alpha<1/2$.