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Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains. (English) Zbl 1341.35098

The Stokes resolvent system \(\lambda \mathbf u -\Delta \mathbf u +\nabla p=\mathbf f\), div \(\mathbf u=0\) in an unbounded domain \(\Omega \subset \mathbb R^n\) (\(n \geq 2\)) is considered. Here, \(\lambda \in \mathbb C \backslash \{0\}\) is contained in the sector \(|\arg \lambda|<\frac{\pi}2+\varepsilon\), \(0<\varepsilon<\frac{\pi}2\). The boundary conditions \(\mathbf u \cdot \mathbf n =0\) and \(\alpha \mathbf u +\beta \mathbf T \mathbf u =0\) (Navier’s condition) are posed on \(\partial \Omega \subset C^3\). Let \(A\) be the Stokes operator corresponding to a Helmholtz projection \(P\). The authors investigate the resolvent problem \(\lambda \mathbf u+A \mathbf u = P \mathbf f\) in the space \(L^q \cap L^2\) when \(q \geq 2\) and in \(L^q + L^2\) when \(1<q<2\).

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
47A10 Spectrum, resolvent
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Full Text: Euclid