# zbMATH — the first resource for mathematics

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
Full Text: