Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908598116810752 |
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| author | Geng, Jun Shen, Zhongwei |
| author_facet | Geng, Jun Shen, Zhongwei |
| contents | We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $Ω$ in $R^d$ under the assumptions that $Ω$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $Ω$. The smoothness conditions on $Ω$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03844 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains Geng, Jun Shen, Zhongwei Analysis of PDEs We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $Ω$ in $R^d$ under the assumptions that $Ω$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $Ω$. The smoothness conditions on $Ω$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level. |
| title | Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.03844 |