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\(L^q\)-approach to weak solutions of the Oseen flow around a rotating body. (English) Zbl 1148.76017

Rencławowicz, Joanna (ed.) et al., Parabolic and Navier-Stokes equations. Part 1. Proceedings of the confererence, Bȩdlewo, Poland, September 10–17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 259-276 (2008).
Summary: We consider time-periodic Oseen flow around a rotating body in \({\mathbb R}^3\). We prove a priori estimates in \(L^q\)-spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates, the problem is reduced to a stationary Oseen equation with the additional term \(-(\omega \wedge x) \cdot \nabla u + \omega \wedge u\) in the equation of momentum, where \(\omega\) denotes the angular velocity. We prove the existence of generalized weak solutions in \(L^q\)-space using Littlewood-Paley decomposition and maximal operators.
For the entire collection see [Zbl 1147.35005].

MSC:

76D07 Stokes and related (Oseen, etc.) flows
76U05 General theory of rotating fluids
35Q35 PDEs in connection with fluid mechanics
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