On Stokes Operators with Variable Viscosity in Bounded and Unbounded Domains

Helmut Abels and Yutaka Terasawa

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Submission date: 25. Mar. 2008 (revised version: October 2008)
Pages: 50
published in: Mathematische Annalen, 344 (2009) 2, p. 381-429 
DOI number (of the published article): 10.1007/s00208-008-0311-7
Bibtex
MSC-Numbers: 35Q30, 76D07, 47A60, 47F05
Keywords and phrases: Stokes operator, Stokes equation, unbounded domains, bounded imaginary powers, $H_\infty$-calculus
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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 -calculus, which implies the maximal -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with -boundary for some r>d with . In particular, the existence of an -Helmholtz projection is assumed.