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Bounded $ H^{\infty}$-calculus for the hydrostatic Stokes operator on $ L^p$-spaces and applications


Authors: Yoshikazu Giga, Mathis Gries, Matthias Hieber, Amru Hussein and Takahito Kashiwabara
Journal: Proc. Amer. Math. Soc. 145 (2017), 3865-3876
MSC (2010): Primary 35Q35; Secondary 47D06, 76D03
DOI: https://doi.org/10.1090/proc/13676
Published electronically: May 24, 2017
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Abstract: It is shown that the hydrostatic Stokes operator on $ L^p_{\overline {\sigma }}(\Omega )$, where $ \Omega \subset \mathbb{R}^3$ is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded $ H^\infty $-calculus on $ L^p_{\overline {\sigma }}(\Omega )$ for $ p\in (1,\infty )$ of $ H^\infty $-angle 0. In particular, maximal $ L^q-L^p$-regularity estimates for the linearized primitive equations are obtained.


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Additional Information

Yoshikazu Giga
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
Email: labgiga@ms.u-tokyo.ac.jp

Mathis Gries
Affiliation: Departement of Mathematics, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email: gries@mathematik.tu-darmstadt.de

Matthias Hieber
Affiliation: Departement of Mathematics, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email: hieber@mathematik.tu-darmstadt.de

Amru Hussein
Affiliation: Departement of Mathematics, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email: hussein@mathematik.tu-darmstadt.de

Takahito Kashiwabara
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
Email: tkashiwa@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/proc/13676
Keywords: Hydrostatic Stokes operator, $H^{\infty}$-functional calculus, maximal $L^p$-regularity
Received by editor(s): July 1, 2016
Received by editor(s) in revised form: August 2, 2016
Published electronically: May 24, 2017
Additional Notes: This work was partly supported by the DFG International Research Training Group IRTG 1529 and the JSPS Japanese-German Graduate Externship on Mathematical Fluid Dynamics.
The first author is partly supported by JSPS through grant Kiban S (No. 26220702).
The second and fourth authors are supported by IRTG 1529 at TU Darmstadt
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society

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