Koba, Hajime On stability of Leray’s stationary solutions of the Navier-Stokes system in exterior domains. (English) Zbl 1368.35208 J. Math. Soc. Japan 69, No. 1, 373-396 (2017). 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. Reviewer: Gheorghe Moroşanu (Budapest) Cited in 1 Document MSC: 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 93D20 Asymptotic stability in control theory Keywords:Navier-Stokes in exterior domains; asymptotic stability; decay property; maximal \(L^p\) regularity PDFBibTeX XMLCite \textit{H. Koba}, J. Math. Soc. Japan 69, No. 1, 373--396 (2017; Zbl 1368.35208) Full Text: DOI Euclid