Full instability of boundary layers with the Navier boundary condition
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909060676190208 |
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| author | Quarisa, Lorenzo Rodrigo, José L. |
| author_facet | Quarisa, Lorenzo Rodrigo, José L. |
| contents | We consider the problem of the stability of the Navier-Stokes equations in $\mathbb{T}\times \mathbb{R}_+$ near shear flows which are linearly unstable for the Euler equation. In \cite{greniernguyen}, the authors prove an $L^{\infty}$ instability result for the no-slip boundary condition which also denies the validity of the Prandtl boundary layer expansion. In this paper, we generalise this result to a Navier slip boundary condition with viscosity dependent slip length: $\partial_y u =ν^{-γ}u$ at $y=0$, where $γ>1/2$. This range includes the physical slip rate $γ=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_14999 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Full instability of boundary layers with the Navier boundary condition Quarisa, Lorenzo Rodrigo, José L. Analysis of PDEs We consider the problem of the stability of the Navier-Stokes equations in $\mathbb{T}\times \mathbb{R}_+$ near shear flows which are linearly unstable for the Euler equation. In \cite{greniernguyen}, the authors prove an $L^{\infty}$ instability result for the no-slip boundary condition which also denies the validity of the Prandtl boundary layer expansion. In this paper, we generalise this result to a Navier slip boundary condition with viscosity dependent slip length: $\partial_y u =ν^{-γ}u$ at $y=0$, where $γ>1/2$. This range includes the physical slip rate $γ=1$. |
| title | Full instability of boundary layers with the Navier boundary condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.14999 |