Ann Univ Ferrara (2009) 55:199–207 DOI 10.1007/s11565-009-0074-4 Navier–Stokes equations on exterior domains: review for the stability for the incompressible flow on exterior domains Hyeong-Ohk Bae Received: 28 January 2009 / Accepted: 1 July 2009 / Published online: 19 September 2009 © Università degli Studi di Ferrara 2009 Abstract We review some results on the stability of the incompressible fluids in an exterior domain. We categorize the survey according to the values of the steady solutions and to the values of the far field velocity, and to the spatial dimension. Keywords Navier–Stokes equations · Temporal stability · Weighted stability · Exterior domain · Wake, exterior domain · Decay · Steady solution · Weak solution · Strong solution Mathematics Subject Classification (2000) 35Q30 · 76D05 The exterior problem has many applications, for example, sailing boats, flying air- planes or golf balls, hemoglobin in blood, moving shark, a cilium in the sea. In general, these objects sometimes rotate and move forward and backward, which cause the flow. In such case it is not so easy to analyze the motion. As a first approximation people assume that such objects do not rotate, and move with a constant velocity. When a boat is sailing with constant velocity b, we may think that the water is flowing around the fixed boat with opposite velocity -b like the water flow around an island. As we have seen, behind the boat there is a parabolic region, in which the motion of the water is significantly different from that in the other area. Such parabolic region is called the wake. In Galdi’s series books [15, 16] such a phenomenon for the steady case is explained well. The mathematical background is also well described in This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00020). H.-O. Bae (B ) Department of Mathematics, Ajou University, Suwon 443-749, Korea e-mail: hobae@ajou.ac.kr 123