On stability of Leray's stationary solutions of the Navier-Stokes system in exterior domains (Q520645)

Let \(\Omega \subset \mathbb{R}^3\) be an exterior domain with smooth boundary \(\partial \Omega\). Consider the system \[ \begin{aligned} & u_t-\nu\Delta u+(u,\nabla)u+\nabla\Pi=\nabla \cdot F \;\text{ in } \Omega \times (0, \infty) , \\ & \nabla \cdot u =0 \;\text{ in } \Omega \times (0, \infty), \\ & u|_{\partial \Omega}=0, \;\lim_{|x|\rightarrow \infty} u=u_{\infty}, \;u|_{t=0}=u_0, \end{aligned} \] where the notations are the usual ones. The author studies the stability of the solution of the above system around the stationary solution \((w, \pi)\), i.e., the solution of the system \[ \begin{aligned} &-\nu \Delta w +(u_{\infty}, \nabla)w+(w,\nabla)w+\nabla \pi =\nabla \cdot F \;\text{ in } \Omega, \\ & \nabla \cdot w=0 \;\text{ in } \Omega, \\ & w|_{\partial \Omega}=-u_{\infty}, \;\lim_{|x|\rightarrow \infty} w=0, \end{aligned} \] which satisfies \(\int_{\Omega}|\nabla w(x)|^2dx < \infty\) (i.e., it is a Leray stationary solution). The main results are contained in Theorem 2.1, where \(\| w\|_{L^{3,\infty}(\Omega)}\) is assumed to be small.