Navier–Stokes equations on exterior domains: review for the stability for the incompressible flow on exterior domains

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.The exterior problem has many applications, for example, sailing boats, flying airplanes 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