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arXiv · 2401.14642

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

Yanqiu Guo
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Abstract

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 $\nabla \cdot u=0$, for any $β> \frac{17}{12} $. The exponent $β= \frac{3}{2}$ 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 $β< \frac{3}{2}$. An important aspect of our argument involves a refined analysis on the sparse distribution of lattice points in annular regions.

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Yanqiu Guo. 2024-01-26. Inertial manifolds for the two-dimensional hyperviscous Navier-Stokes equations. https://arxiv.org/abs/2401.14642

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