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Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$

Zhile Li

TL;DR

This work proves stability of the 2D Navier–Stokes flow near the Poiseuille profile on $\mathbb{R}^2$. It combines a linear enhanced dissipation analysis, via a tailored energy $E_k$ with frequencies and weights, with a nonlinear framework that uses a time-dependent multiplier $M_k(t)$ to control energy growth. Under small anisotropic perturbations, with norm $\mathcal{E}(0)\lesssim \nu^{7/3}$, the solution remains uniformly bounded and decays, demonstrating quantitative nonlinear stability and a dissipation time-scale faster than the heat equation for high x-frequencies. The approach highlights how linear decay mechanisms extend to the nonlinear regime through carefully designed energies and multi-frequency estimates, contributing to the understanding of shear-flow stability in unbounded domains.

Abstract

We consider solutions to the Navier-Stokes equations on $\mathbb{R}^2$ close to the Poiseuille flow with viscosity $0< ν< 1$. For the linearized problem, we prove that when the $x$-frequency satisfy $|k| \ge ν^{-\frac{1}{3}}$, the perturbation decays on a time-scale proportional to $ν^{-\frac{1}{2}}|k|^{-\frac{1}{2}}$. Since it decays faster than the heat equation, this phenomenon is referred to as enhanced dissipation. Then we concern the non-linear equations. We show that if the initial perturbation $ω_{in}$ is at most of size $ν^\frac{7}{3}$ in an anisotropic Sobolev space, then the size of the perturbation remains no more than twice the size of its initial value.

Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$

TL;DR

This work proves stability of the 2D Navier–Stokes flow near the Poiseuille profile on . It combines a linear enhanced dissipation analysis, via a tailored energy with frequencies and weights, with a nonlinear framework that uses a time-dependent multiplier to control energy growth. Under small anisotropic perturbations, with norm , the solution remains uniformly bounded and decays, demonstrating quantitative nonlinear stability and a dissipation time-scale faster than the heat equation for high x-frequencies. The approach highlights how linear decay mechanisms extend to the nonlinear regime through carefully designed energies and multi-frequency estimates, contributing to the understanding of shear-flow stability in unbounded domains.

Abstract

We consider solutions to the Navier-Stokes equations on close to the Poiseuille flow with viscosity . For the linearized problem, we prove that when the -frequency satisfy , the perturbation decays on a time-scale proportional to . Since it decays faster than the heat equation, this phenomenon is referred to as enhanced dissipation. Then we concern the non-linear equations. We show that if the initial perturbation is at most of size in an anisotropic Sobolev space, then the size of the perturbation remains no more than twice the size of its initial value.

Paper Structure

This paper contains 11 sections, 7 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Linearized Problem
  3. Non-linear Problem
  4. Technical Lemmas
  5. Bound on $T_1$
  6. Bound on $T_2, T_3$ and $T_4$
  7. Bound on $T_5$
  8. Bound on $T_6$
  9. Bound on $T_7$
  10. Bound on $T_8$
  11. Bound on $\mathbb{NL}$

Key Result

Theorem 1.1

Suppose $\omega_{in}$ is initial datum for eq 1.3. Then for all $J \in [1, +\infty), m \in (\frac{3}{4}, 1)$, there exists a constant $\delta > 0$ independent of $\nu$ such that if then for all $c > 0$ sufficiently small (independent of $\nu$ and $\delta$) and all $\nu \in (0, 1)$, the corresponding solution $\omega$ to eq 1.3 satisfies Here $\lambda_\nu^{pl}(\partial_x)$ is a Fourier multiplier

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof
  • proof
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 8 more