On stability of Leray’s stationary solutions of the Navier-Stokes system in exterior domains. (English) Zbl 1368.35208

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.


35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
93D20 Asymptotic stability in control theory
Full Text: DOI Euclid