Functions and Analysis

    

Inertial Manifold for Dyadic Model of the Navier-Stokes Equations

Authors: Yuhan Wei

We prove the existence of finite-dimensional inertial manifolds forthe dyadic model of turbulence for all dissipation exponents α≥1/3.For α = 1/3 and α > 1/3 the proof is unified by working in the Hα-norm and employing a generalized cone method. The dimension scales as N∼1 2αlog λ log ν−1, matching the optimal upper bounds for shell models. The construction relies on a low-mode cut-off, a forward cascade estimate that exploits the monotone structure of the dyadic model, and a modified strong squeezing property of Koksch (2000). The resulting inertial manifold is Lipschitz and C1+ϵ-smooth, and satisfies the exponential tracking property. This provides a rigorousfinite-dimensional reduction for the entire supercritical range α≥1/3. We also answer an open question by Cheskidov (2008) regarding theexistence of strong compact global attractors for α<1/2.

Comments: 35 Pages.

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[v1] 2026-06-29 20:37:01

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