Inertial manifolds for the two-dimensional hyperviscous Navier-Stokes equations

This study establishes the existence of inertial manifolds for the hyperviscous Navier-Stokes equations (HNSE) on a 2D periodic domain: \begin{equation*} \partial_t u+ ν(-Δ) ^βu+(u\cdot \nabla )u+\nabla p=f, ;; $\text{on}$ ;; $\mathbb{T}$^2, \end{equation*} with $\mathrm{\nabla }\cdot u=0$, for any $\beta >\frac{17}{12}$. The exponent $\beta =\frac{3}{2}$ is identified as the “critical” value for the inertial manifold problem in 2D HNSE, below which the spectral gap condition is not satisfied. A breakthrough in this work is that it extends the theory to “supercritical” regimes where $\beta <\frac{3}{2}$. An important aspect of our argument involves a refined analysis on the sparse distribution of lattice points in annular regions.