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An $$L^q(L^2)$$-theory of the generalized Stokes resolvent system in infinite cylinders. (English) Zbl 1111.35034
The authors study the generalized Stokes resolvent system $$\lambda u - \Delta u +\nabla p =f,\; \;div\; u=g$$ in an infinite cylinder $$\Sigma\times \mathbb R$$ with the Dirichlet boundary condition. They prove apriori estimates for $$u, \nabla^2 u, \nabla p$$ in the metric $$L^q(\mathbb R;L^2(\Sigma))$$, $$1<q<\infty,$$ if the spectral parameter $$\lambda$$ lies in a sector with vertex $$-\alpha$$, where $$0<\alpha<\alpha_0$$ and $$\alpha_0$$ is the smallest eigenvalue of the Dirichlet Laplacian in $$\Sigma.$$

##### MSC:
 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows 37C27 Periodic orbits of vector fields and flows 42A45 Multipliers in one variable harmonic analysis 46E40 Spaces of vector- and operator-valued functions
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