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

Yanqiu Guo

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.

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

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 , for any . The exponent 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 . An important aspect of our argument involves a refined analysis on the sparse distribution of lattice points in annular regions.
Paper Structure (10 sections, 8 theorems, 74 equations)

This paper contains 10 sections, 8 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. The literature
  4. The main result
  5. The proof of the main result
  6. An abstract model
  7. Sparse distribution of lattice points in annuli
  8. Modification of the HNSE outside the absorbing ball
  9. Verification of the spatial averaging condition
  10. Discussion

Key Result

Theorem 2.2

Consider the Hilbert space $\mathbb H$ defined in (H-space). Assume $\beta > \frac{17}{12}$. Let $f\in H^{\frac{1}{6}}(\mathbb T^2)$. Then the "prepared" equation (prepared) for the hyperviscous Navier-Stokes equations (HNSEb) has an inertial manifold in $\mathbb H$ in the sense of Definition defman

Theorems & Definitions (18)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Remark 3.5
  • ...and 8 more