## 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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