## 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.

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 93D20 Asymptotic stability in control theory
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