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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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