Transition threshold of Couette flow for 2D Boussinesq equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912412907601920 |
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| author | Ren, Xiaoxia Dongyi, Wei |
| author_facet | Ren, Xiaoxia Dongyi, Wei |
| contents | In this paper, we prove the stability threshold of $α\leq \frac13$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number $γ^2>\frac14$ and different viscosity $ν$ and thermal diffusivity $μ$. More precisely, if $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|ρ_{in}+γ^2 y-1\|_{H^{s+1/2}}\leq c(\min\{ν,μ\})^{1/3}$, $\frac{ν+μ}{2γ\sqrt{νμ} }< 2-\varepsilon$, $s>3/2$, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transition threshold of Couette flow for 2D Boussinesq equations Ren, Xiaoxia Dongyi, Wei Analysis of PDEs In this paper, we prove the stability threshold of $α\leq \frac13$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number $γ^2>\frac14$ and different viscosity $ν$ and thermal diffusivity $μ$. More precisely, if $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|ρ_{in}+γ^2 y-1\|_{H^{s+1/2}}\leq c(\min\{ν,μ\})^{1/3}$, $\frac{ν+μ}{2γ\sqrt{νμ} }< 2-\varepsilon$, $s>3/2$, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp. |
| title | Transition threshold of Couette flow for 2D Boussinesq equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.03679 |