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Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body. (English) Zbl 1234.35035
Summary: Consider a body, \(\mathcal B\), rotating with constant angular velocity \(\omega \) and fully submerged in a Navier-Stokes liquid that fills the whole space exterior to \(\mathcal B\). We analyze the flow of the liquid that is steady with respect to a frame attached to \(\mathcal B\). Our main theorem shows that the velocity field \(v\) of any weak solution \((v,p)\) in the sense of Leray has an asymptotic expansion with a suitable Landau solution as leading term and a remainder decaying pointwise like \(1/|x|^{1+\alpha }\) as \(|x|\rightarrow \infty \) for any \(\alpha \in (0,1)\), provided the magnitude of \(\omega \) is below a positive constant depending on \(\alpha \). We also furnish analogous expansions for \(\nabla v\) and for the corresponding pressure field \(p\). These results improve and clarify a recent result of the first author and T. Hishida [Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 55, No. 2, 263–277 (2009; Zbl 1205.35191); Math. Nachr. 284, No. 16, 2065–2077 (2011; Zbl 1229.35173)].

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
76U05 General theory of rotating fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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