On viscous incompressible flows around a rotating obstacle in two dimensions

On viscous incompressible flows around a rotating obstacle in two dimensions

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Yasunori Maekawa , Tohoku University
Fine Hall 322

In this talk we consider the Navier-Stokes equations for viscous incompressible flows in a two-dimensional exterior domain subject to the no-slip boundary condition. Due to the boundary condition the motion of the obstacle (complement of the exterior domain) naturally create a nontrivial flow. The most typical case is that the obstacle is translating with a constant velocity, known as the Oseen problem, and the stationary Navier-Stokes flows in this case were obtained by Finn and Smith in 1960's. Another typical motion of the obstacle is the rotation with a constant angular velocity, however, the existence of the corresponding stationary Navier-Stokes flows (in the reference frame) has been open in the two-dimensional case. Recently, by analyzing the linearized problem, Hishida (2015) revealed that the rotation of the obstacle resolves the Stokes paradox as in the Oseen case. In this talk we prove the existence of stationary Navier-Stokes flows by extending Hishida's result for the linearized problem. The asymptotic behavior of solutions at spatial infinity is also clarified. The stability of these stationary solutions is a difficult issue and will be presented only for a special case. A part of this talk is based on the collaboration with Mitsuo Higaki (Tohoku University) and Yuu Nakahara (Tohoku University).