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This article concentrates on various operator semigroups arising in the study of viscous and incompressible flows. Of particular concern are the classical Stokes semigroup, the hydrostatic Stokes semigroup, the Oldroyd as well as the Ericksen-Leslie semigroup. Besides their intrinsic interest, the properties of these semigroups play an important role in the investigation of the associated nonlinear equations. This article is part of the theme issue 'Semigroup applications everywhere'.In this paper, we introduce a general framework to stu
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