Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916338350424064 |
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| author | Kang, Kyungkeun Lee, Jihoon Nguyen, Dinh Duong |
| author_facet | Kang, Kyungkeun Lee, Jihoon Nguyen, Dinh Duong |
| contents | The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to $H^δ(\mathbb{R}^2) \times H^{s}(\mathbb{R}^2)$ for $δ\in [s-1,s+1]$ if $s \in \mathbb{R}, s > 2$, for $δ\in (1,s+1]$ if $s \in (0,2]$ and for $δ\in [0,1]$ if $s = 0$. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in $H^m(\mathbb{R}^2)$, $m \in \mathbb{N}$ and in $\dot{H}^{m-1}(\mathbb{R}^2)$ with $m \in \mathbb{N}$, $m \geq 2$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08286 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium Kang, Kyungkeun Lee, Jihoon Nguyen, Dinh Duong Analysis of PDEs 35B35, 35Q35, 76D03 The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to $H^δ(\mathbb{R}^2) \times H^{s}(\mathbb{R}^2)$ for $δ\in [s-1,s+1]$ if $s \in \mathbb{R}, s > 2$, for $δ\in (1,s+1]$ if $s \in (0,2]$ and for $δ\in [0,1]$ if $s = 0$. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in $H^m(\mathbb{R}^2)$, $m \in \mathbb{N}$ and in $\dot{H}^{m-1}(\mathbb{R}^2)$ with $m \in \mathbb{N}$, $m \geq 2$, respectively. |
| title | Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium |
| topic | Analysis of PDEs 35B35, 35Q35, 76D03 |
| url | https://arxiv.org/abs/2306.08286 |