Stability of the Couette flow under the 2D steady Navier-Stokes flow. (English) Zbl 1527.35221
Summary: In this note, we investigate the stability property of shear flows under the 2D stationary Navier-Stokes equations, and obtain that the Couette flow \((y, 0)\) is stable under the space of \(\mathcal{D}^{1,q}(\mathbb{R}^2)\) for any \(1<q<\infty\) and unstable in the space of \(\mathcal{D}^{1,\infty}(\mathbb{R}^2)\), which is sharp in this sense. A key observation is the choice of the anisotropic cut-off function. The Poiseuille flow \((y^2, 0)\) is also considered as a by-product, which is stable in the space of \(\mathcal{D}^{1,q}(\mathbb{R}^2)\) with \(\frac{4}{3}<q\leq 4\) via a lemma of Fefferman-Stein.
{© 2022 Wiley-VCH GmbH.}
{© 2022 Wiley-VCH GmbH.}
MSC:
| 35Q30 | Navier-Stokes equations |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B35 | Stability in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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