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.
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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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Abels, H., Terasawa, Y. On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344, 381–429 (2009). https://doi.org/10.1007/s00208-008-0311-7
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DOI: https://doi.org/10.1007/s00208-008-0311-7