A dynamical approach to the study of instability near Couette flow
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913333828911104 |
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| author | Li, Hui Masmoudi, Nader Zhao, Weiren |
| author_facet | Li, Hui Masmoudi, Nader Zhao, Weiren |
| contents | In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $ν>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_10894 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A dynamical approach to the study of instability near Couette flow Li, Hui Masmoudi, Nader Zhao, Weiren Analysis of PDEs Fluid Dynamics In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $ν>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues. |
| title | A dynamical approach to the study of instability near Couette flow |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2203.10894 |