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MiS Preprint

On Stokes Operators with Variable Viscosity in Bounded and Unbounded Domains

Helmut Abels and Yutaka Terasawa


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 $H_\infty$-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 $r\geq q$. In particular, the existence of an $L^q$-Helmholtz projection is assumed.

MSC Codes:
35Q30, 76D07, 47A60, 47F05
Stokes operator, Stokes equation, unbounded domains, bounded imaginary powers, $H_\infty$-calculus

Related publications

2009 Journal Open Access
Helmut Abels and Yutaka Terasawa

On Stokes operators with variable viscosity in bounded and unbounded domains

In: Mathematische Annalen, 344 (2009) 2, pp. 381-429