$H^\infty$-calculus for Stokes operators on rough and on unbounded domains
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| Format: | Preprint |
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2025
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| _version_ | 1866916976393191424 |
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| author | Kunstmann, Peer Christian Tolksdorf, Patrick |
| author_facet | Kunstmann, Peer Christian Tolksdorf, Patrick |
| contents | In this article, we give an overview on known as well as new results on the boundedness of the $H^{\infty}$-calculus of the Stokes operator in rough as well as in unbounded (smoother) domains. We present a special case of an abstract comparison principle due to Kunstmann and Weis (\cite{KuW:Hinfty-Stokes}) that serves as the basis for all considerations. Subsequently, we show how this result can be applied to arrive at a bounded $H^{\infty}$-calculus for the Stokes operator. We sketch the proof for no slip boundary conditions in bounded Lipschitz domains which was given in~\cite{KuW:Hinfty-Stokes}. For unbounded domains this approach yields a shorter proof compared to previous arguments. Moreover, we further establish the boundedness of the $H^{\infty}$-calculus for the Stokes operator with Neumann type boundary conditions in bounded convex domains which is entirely new. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $H^\infty$-calculus for Stokes operators on rough and on unbounded domains Kunstmann, Peer Christian Tolksdorf, Patrick Analysis of PDEs Functional Analysis 35Q30, 76D07, 47A60, 47D06 In this article, we give an overview on known as well as new results on the boundedness of the $H^{\infty}$-calculus of the Stokes operator in rough as well as in unbounded (smoother) domains. We present a special case of an abstract comparison principle due to Kunstmann and Weis (\cite{KuW:Hinfty-Stokes}) that serves as the basis for all considerations. Subsequently, we show how this result can be applied to arrive at a bounded $H^{\infty}$-calculus for the Stokes operator. We sketch the proof for no slip boundary conditions in bounded Lipschitz domains which was given in~\cite{KuW:Hinfty-Stokes}. For unbounded domains this approach yields a shorter proof compared to previous arguments. Moreover, we further establish the boundedness of the $H^{\infty}$-calculus for the Stokes operator with Neumann type boundary conditions in bounded convex domains which is entirely new. |
| title | $H^\infty$-calculus for Stokes operators on rough and on unbounded domains |
| topic | Analysis of PDEs Functional Analysis 35Q30, 76D07, 47A60, 47D06 |
| url | https://arxiv.org/abs/2509.24501 |