Stability threshold of Couette flow for Boussinesq equations in $\mathbb{R}^2$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916902589169664 |
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| author | Chen, Yubo Wang, Wendong Yang, Guoxu |
| author_facet | Chen, Yubo Wang, Wendong Yang, Guoxu |
| contents | This paper establishes the asymptotic stability threshold for the Couette flow $(y,0)$ under the 2D Boussinesq system in $\mathbb{R}^2$. It was proved that for initial perturbations in Sobolev spaces with controlled low horizontal frequencies, the stability threshold is at most $\left\{\frac{1}{3}+, \frac{2}{3}+\right\}$, extending the known threshold results from the periodic case $\mathbb{T}_x \times \mathbb{R}_y$ to the whole space. The core innovations are twofold: First, the $\langle D_x^{-1} \rangle$ control on the initial data simultaneously resolves horizontal frequency singularities and optimizes integral indices when applying Young's convolution inequality. Second, we develop a modified multiplier $\mathcal{M}_3$ that effectively absorbs the $|D_x|^{1/3}$ derivative structure induced by the temperature equation while handling nonlinear echo cascades. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability threshold of Couette flow for Boussinesq equations in $\mathbb{R}^2$ Chen, Yubo Wang, Wendong Yang, Guoxu Analysis of PDEs This paper establishes the asymptotic stability threshold for the Couette flow $(y,0)$ under the 2D Boussinesq system in $\mathbb{R}^2$. It was proved that for initial perturbations in Sobolev spaces with controlled low horizontal frequencies, the stability threshold is at most $\left\{\frac{1}{3}+, \frac{2}{3}+\right\}$, extending the known threshold results from the periodic case $\mathbb{T}_x \times \mathbb{R}_y$ to the whole space. The core innovations are twofold: First, the $\langle D_x^{-1} \rangle$ control on the initial data simultaneously resolves horizontal frequency singularities and optimizes integral indices when applying Young's convolution inequality. Second, we develop a modified multiplier $\mathcal{M}_3$ that effectively absorbs the $|D_x|^{1/3}$ derivative structure induced by the temperature equation while handling nonlinear echo cascades. |
| title | Stability threshold of Couette flow for Boussinesq equations in $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.11908 |