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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 \(\widetilde{L}^q(\Omega) = L^2(\Omega)+L^q(\Omega),~ 1<q<2\).

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