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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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