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arXiv · 2411.19716

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

Zhile Li
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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.

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Zhile Li. 2025-03-23. Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$. https://arxiv.org/abs/2411.19716

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