Nonlinear stability and transition threshold for the planar helical flow
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| Format: | Preprint |
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2024
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| _version_ | 1866909262729445376 |
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| author | Shi, Binbin Wang, Yucheng |
| author_facet | Shi, Binbin Wang, Yucheng |
| contents | In this paper, we study the nonlinear stability for the 3-D planar helical flow $(δ^2\sin(m_0 y),δ^2\cos(m_0 y),0)$ on torus $\mathbb{T}^3=\{(x_1,x_2,y)\big|x_1,x_2\in \mathbb{T}_{2π}, y\in \mathbb{T}_{2πδ}, δ\geq1\}$ for high Reynolds number $Re$. We prove that if the initial velocity $U_0$ satisfies $$ \left\|U_0-(δ^2\sin(m_0 y),δ^2\cos(m_0 y),0)\right\|_{X_0}\leq c_0 Re^{-7/4} $$ for some $c_0>0$ independent of $Re$, then the solution of 3-D incompressible Navier-Stokes equation is global in time and does not transit away from the planar helical flow. Here $δ>1, m_0=δ^{-1}$ and the norm $\|\cdot\|_{X_0}$ is defined in (1.8). This is a nonlinear stability result for 3-D non-shear flow and the transition threshold is less than $7/4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_11298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear stability and transition threshold for the planar helical flow Shi, Binbin Wang, Yucheng Analysis of PDEs 35Q30, 35Q35, 76E06 In this paper, we study the nonlinear stability for the 3-D planar helical flow $(δ^2\sin(m_0 y),δ^2\cos(m_0 y),0)$ on torus $\mathbb{T}^3=\{(x_1,x_2,y)\big|x_1,x_2\in \mathbb{T}_{2π}, y\in \mathbb{T}_{2πδ}, δ\geq1\}$ for high Reynolds number $Re$. We prove that if the initial velocity $U_0$ satisfies $$ \left\|U_0-(δ^2\sin(m_0 y),δ^2\cos(m_0 y),0)\right\|_{X_0}\leq c_0 Re^{-7/4} $$ for some $c_0>0$ independent of $Re$, then the solution of 3-D incompressible Navier-Stokes equation is global in time and does not transit away from the planar helical flow. Here $δ>1, m_0=δ^{-1}$ and the norm $\|\cdot\|_{X_0}$ is defined in (1.8). This is a nonlinear stability result for 3-D non-shear flow and the transition threshold is less than $7/4$. |
| title | Nonlinear stability and transition threshold for the planar helical flow |
| topic | Analysis of PDEs 35Q30, 35Q35, 76E06 |
| url | https://arxiv.org/abs/2404.11298 |