Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel
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arXiv
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| Format: | Preprint |
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2025
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| author | Liang, Tao Wu, Jiahong Zhai, Xiaoping |
| author_facet | Liang, Tao Wu, Jiahong Zhai, Xiaoping |
| contents | In this paper, we study the stability threshold of the two-dimensional Boussinesq equations around the Couette flow in an infinite channel $\mathbb{R} \times [-1, 1]$ under no-slip boundary conditions. We prove that the Couette flow is asymptotically stable under initial perturbations satisfying $\| \mathbf{v}^{\mathrm{in}} -(y,0)\|_{H^2} \le \varepsilon_0 ν^{\frac12}$, and $\| ρ^{\mathrm{in}}-1 \|_{H^1} + \big\| |\partial_x|^{\frac13} ρ^{\mathrm{in}} \big\|_{H^1} \le \varepsilon_1 ν^{\frac56}$. Compared with the work of Masmoudi, Zhai, and Zhao [J. Funct. Anal., 284 (2023), 109736], where the asymptotic stability of the 2D Navier-Stokes-Boussinesq system around Couette flow in a finite channel $\mathbb{T} \times [-1, 1]$ was established, our result improves the stability threshold for the temperature from $ν^{\frac{11}{12}}$ to $ν^{\frac56}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel Liang, Tao Wu, Jiahong Zhai, Xiaoping Analysis of PDEs In this paper, we study the stability threshold of the two-dimensional Boussinesq equations around the Couette flow in an infinite channel $\mathbb{R} \times [-1, 1]$ under no-slip boundary conditions. We prove that the Couette flow is asymptotically stable under initial perturbations satisfying $\| \mathbf{v}^{\mathrm{in}} -(y,0)\|_{H^2} \le \varepsilon_0 ν^{\frac12}$, and $\| ρ^{\mathrm{in}}-1 \|_{H^1} + \big\| |\partial_x|^{\frac13} ρ^{\mathrm{in}} \big\|_{H^1} \le \varepsilon_1 ν^{\frac56}$. Compared with the work of Masmoudi, Zhai, and Zhao [J. Funct. Anal., 284 (2023), 109736], where the asymptotic stability of the 2D Navier-Stokes-Boussinesq system around Couette flow in a finite channel $\mathbb{T} \times [-1, 1]$ was established, our result improves the stability threshold for the temperature from $ν^{\frac{11}{12}}$ to $ν^{\frac56}$. |
| title | Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.01159 |