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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 \(\Omega\).

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