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Abstract
In this paper we develop a functional framework for unsteady problems for monotone operators in domains with smoothly moving boundaries. This is motivated by the motion of generalized Newtonian fluids in such domains. A crucial step towards this is the proof of a general integration by-parts formula for functions in non-cylindrical domains. The framework is used to show the existence of weak solutions for the p-Stokes problem in non-cylindrical domains.
Notes
No potential conflict of interest was reported by the authors.
1 We use the conventional notation from integration theory.
2 We use the convenient notation to denote the set of
-functions defined on some open subset
if this set is equipped with the usual topology. By
we mean the set of
-fields. A continuous linear mapping
is called distribution and the set of distributions on U is denoted by
. The set
is defined along the lines.
3 Of course we assume that the integrals exist. This can be guaranteed through restrictions on the respective exponents of integrability. However, we suppress these restrictions in this definition for the sake of readability.
4 with the convention that is extended by
to