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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04636 |
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| _version_ | 1866916197258231808 |
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| author | Zhang, Huiyang Yan, Shuokai Zhang, Qinghua |
| author_facet | Zhang, Huiyang Yan, Shuokai Zhang, Qinghua |
| contents | This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-Δ)^α$ in $\mathbb{R}^{n}$ for $n\geq3$. It proves the existence of a small positive number $\varepsilon=\varepsilon(n,α)$ such that for each $0<T<\infty$, if $\frac{1}{2}<α<\frac{2+n}{4}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+T^{1/2}\|θ_{0}\|_{\dot{H}^{s_{0}-α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the bounded interval $[0,T]$. If $\frac{1}{2}<α<\frac{2+n}{6}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+\|θ_{0}\|_{\dot{H}^{s_{0}-2α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the whole interval $[0,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global solvability for the Boussinesq system with fractional Laplacian Zhang, Huiyang Yan, Shuokai Zhang, Qinghua Analysis of PDEs 35Q30, 76D05 This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-Δ)^α$ in $\mathbb{R}^{n}$ for $n\geq3$. It proves the existence of a small positive number $\varepsilon=\varepsilon(n,α)$ such that for each $0<T<\infty$, if $\frac{1}{2}<α<\frac{2+n}{4}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+T^{1/2}\|θ_{0}\|_{\dot{H}^{s_{0}-α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the bounded interval $[0,T]$. If $\frac{1}{2}<α<\frac{2+n}{6}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+\|θ_{0}\|_{\dot{H}^{s_{0}-2α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the whole interval $[0,\infty)$. |
| title | Global solvability for the Boussinesq system with fractional Laplacian |
| topic | Analysis of PDEs 35Q30, 76D05 |
| url | https://arxiv.org/abs/2404.04636 |