On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space
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arXiv
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| Format: | Preprint |
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2023
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| author | Pražák, Dalibor Zelina, Michael |
| author_facet | Pražák, Dalibor Zelina, Michael |
| contents | We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a $d$-dimensional half-space. Using the standard technique of the Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. $W^{1,p}$) and strong (hence $W^{2,p}$) solutions, which are optimal in terms of the data belonging to appropriate negative Sobolev or fractional Besov space. In the latter case $L^p$-integrability of the pressure gradient is included. We allow for solutions with non-zero divergence, thus preparing the way for extensions to general domains. As a by-product, we show that the system generates an analytic semigroup in $L^p(Ω)\times L^p(\partial Ω)$.
Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal{H}^{\infty}$-calculus are implicitly present; but we stay away from the concept of $R$-boundedness and related heavy functional analytic machinery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04478 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space Pražák, Dalibor Zelina, Michael Analysis of PDEs 76D07, 47D03, 35B65 We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a $d$-dimensional half-space. Using the standard technique of the Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. $W^{1,p}$) and strong (hence $W^{2,p}$) solutions, which are optimal in terms of the data belonging to appropriate negative Sobolev or fractional Besov space. In the latter case $L^p$-integrability of the pressure gradient is included. We allow for solutions with non-zero divergence, thus preparing the way for extensions to general domains. As a by-product, we show that the system generates an analytic semigroup in $L^p(Ω)\times L^p(\partial Ω)$. Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal{H}^{\infty}$-calculus are implicitly present; but we stay away from the concept of $R$-boundedness and related heavy functional analytic machinery. |
| title | On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space |
| topic | Analysis of PDEs 76D07, 47D03, 35B65 |
| url | https://arxiv.org/abs/2312.04478 |