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

2024/01/26 by Yanqiu Guo, Guo, Yanqiu · 1 citation
Engineering · Mathematics · #35Q30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2401.14642

openalex publication_date 2024/01/26 · openalex created_date 2024/01/30 · openalex updated_date 2026/07/28

Abstract

This study establishes the existence of inertial manifolds for the hyperviscous Navier-Stokes equations (HNSE) on a 2D periodic domain: ∂_t u+ ν(-Δ) ^βu+(u⋅ ∇ )u+∇ p=f, on \mathbbT², with ∇ ⋅ u=0, for any β> (17)/(12) . The exponent β= (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< (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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