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An $$L^q$$-approach to Stokes and Navier-Stokes equations in general domains. (English) Zbl 1111.35033
Throughout this paper, $$\Omega\subseteq {\mathbb R}^3$$ means a general 3-dimensional domain with uniform $$C^2$$- boundary $$\partial\Omega\neq \varnothing$$, where the main interest is focussed on domains with noncompact boundary $$\partial\Omega$$. The standard approach to the Stokes equations in $$L^q$$-spaces, $$1<q<\infty$$, cannot be extended to general unbounded domains in $$L^q$$, $$q\neq2$$; because of counterexamples concerning the Helmholtz decomposition. However, to develop a complete and analogous theory of the Stokes equations for arbitrary domains, the space $$L^q (\Omega)$$ is replaced by $$\widetilde{L}^q(\Omega)=L^2(\Omega)\cap L^q(\Omega),~2 \leq q <\infty,$$ and $$\widetilde{L}^q(\Omega) = L^2(\Omega)+L^q(\Omega),~ 1<q<2$$.

##### MSC:
 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows 76D05 Navier-Stokes equations for incompressible viscous fluids
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