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Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake. (English) Zbl 1119.76011
Summary: This is a sequel to the authors’ paper [Arch. Ration. Mech. Anal. 184, 371–400 (2007; Zbl 1111.76010)].
A rigid body $${\mathcal R}$$ is moving in a Navier-Stokes liquid $${\mathfrak L}$$ that fills the whole space. We assume that all data with respect to a frame, $${\mathcal F}$$, attached to $${\mathcal R}$$, namely, the body force acting on $${\mathfrak L}$$, the boundary conditions on $${\mathcal R}$$ as well as the translational velocity, $$U$$, and the angular velocity, $$\Omega$$, of $${\mathcal R}$$ are independent of time. We assume $$\Omega\neq 0$$ (the case $$\Omega=0$$ being already known) and take, without loss of generality, $$\Omega$$ parallel to the base vector $$e_1$$ in $${\mathcal F}$$. We show that, if the magnitude of these data is not “too large”, there exists at least one steady motion of $${\mathfrak L}$$ in $${\mathcal F}$$, such that the velocity field and its gradient decay like $$(1+|x|)^{-1}(1+2\text{Re}\,s(x))^{-1}$$ and $$(1+|x|)^{-\frac{3}{2}}(1+2\text{Re}\,s(x))^{-\frac {3}{2}}$$, respectively, where Re is the Reynolds number and $$s(x):=|x|+x_1$$ is representative of the “wake” behind the body. This motion is unique in the (larger) class of motions having velocity field decaying like $$|x|^{-1}$$. Since Re is proportional to $$|U\cdot e_1|$$, the above formulas show that the $${\mathfrak L}$$ exhibits a wake behind $${\mathcal R}$$ if and only if $$U$$ is not orthogonal to $$\Omega$$.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D25 Wakes and jets 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
exterior domain; uniqueness; asymptotic behavior