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Helmholtz decomposition and Stokes resolvent system for aperture domains in $$L^ q$$-spaces. (English) Zbl 0847.35101
Summary: In an aperture domain we consider the resolvent problem of the Stokes system in $$L^q$$-spaces. Roughly spoken, an aperture domain $$\Omega\subset \mathbb{R}^n$$, $$n\geq 2$$, is an unbounded domain with noncompact boundary consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall.
Our $$L^q$$-theory allows to prescribe a nonzero divergence of the velocity field and, if $$q> n/(n- 1)$$ requires to prescribe the flux through the hole in order to single out a unique solution. Further we construct the Helmholtz decomposition of $$L^q(\Omega)^n$$ by studying the weak $$L^q$$-theory of the Neumann problem. Then the Stokes operator is shown to generate a bounded analytic semigroup.

MSC:
 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows 47D03 Groups and semigroups of linear operators
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