On Kato's Square Root Property for the Generalized Stokes Operator
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909362904104960 |
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| author | Haardt, Luca Tolksdorf, Patrick |
| author_facet | Haardt, Luca Tolksdorf, Patrick |
| contents | We establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with bounded measurable coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(μ\nabla u) + \nabla ϕ$, $\operatorname{div}(u) = 0$, with the space of divergence-free $\mathrm{H}^1$-vector fields and further prove the estimate $\|A^{1/2} u \|_{\mathrm{L}^2} \simeq \| \nabla u \|_{\mathrm{L}^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $μ$. Besides the boundedness and measurablility as well as an ellipticity condition on $μ$, there are no requirements on the coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Kato's Square Root Property for the Generalized Stokes Operator Haardt, Luca Tolksdorf, Patrick Analysis of PDEs Functional Analysis 47F10, 26A33, 47A60, 35Q35 We establish the Kato square root property for the generalized Stokes operator on $\mathbb{R}^d$ with bounded measurable coefficients. More precisely, we identify the domain of the square root of $Au := - \operatorname{div}(μ\nabla u) + \nabla ϕ$, $\operatorname{div}(u) = 0$, with the space of divergence-free $\mathrm{H}^1$-vector fields and further prove the estimate $\|A^{1/2} u \|_{\mathrm{L}^2} \simeq \| \nabla u \|_{\mathrm{L}^2}$. As an application we show that $A^{1/2}$ depends holomorphically on the coefficients $μ$. Besides the boundedness and measurablility as well as an ellipticity condition on $μ$, there are no requirements on the coefficients. |
| title | On Kato's Square Root Property for the Generalized Stokes Operator |
| topic | Analysis of PDEs Functional Analysis 47F10, 26A33, 47A60, 35Q35 |
| url | https://arxiv.org/abs/2410.18787 |