Stability of Stationary Viscous Incompressible Flow Around a Rigid Body Performing a Translation

Abstract

Suppose a rigid body moves steadily and without rotation in a viscous incompressible fluid, and the flow around the body is steady, too. Such a flow is usually described by the stationary Navier–Stokes system with Oseen term, in an exterior domain. An Oseen term arises because the velocity field is scaled in such a way that it vanishes at infinity. In the work at hand, such a velocity field, denoted by U, is considered as given. We study a solution of the incompressible evolutionary Navier–Stokes system with the same right-hand side and the same Dirichlet boundary conditions as the stationary problem, and with $$U+u_0$$ as initial data, where $$u_0$$ is a $$H^1$$ -function. Under the assumption that the $$H^1$$ -norm of $$u_0$$ is small ( $$u_0$$ a “perturbation of U”) and that the eigenvalues of a certain linear operator have negative real part, we show that $$\Vert \nabla (v(t)-U)\Vert _2\rightarrow 0\; (t\rightarrow \infty )$$ (“stability of v”), where v denotes the velocity part of the solution to the initial-boundary value problem under consideration.