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Mathematische Annalen

, Volume 344, Issue 2, pp 381–429 | Cite as

On Stokes operators with variable viscosity in bounded and unbounded domains

  • Helmut AbelsEmail author
  • Yutaka Terasawa
Open Access
Article

Abstract

We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded H -calculus, which implies the maximal L q -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with \({W^{2-\frac1r}_r}\) -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L q -Helmholtz projection is assumed.

Mathematics Subject Classification (2000)

35Q30 76D07 47A60 47F05 

Notes

Acknowledgments

The authors are grateful to Gerd Grubb and one anonymous referee for several helpful comments to improve the presentation in this contribution. The second author was supported by a research fellowships of the Japan Society for the Promotion of Science for young scientists.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of MathematicsHokkaido UniversitySapporo, HokkaidoJapan

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