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On $$L^1$$-stability of stationary Navier-Stokes flows in $$\mathbb{R}^n$$. (English) Zbl 0990.35117
In the first part of this paper, the author continues his investigations [Kyushu J. Math. 50, 1-64 (1996; Zbl 0883.35088)] for the stationary problem \begin{aligned} & u\cdot\nabla u-\Delta u+\nabla p=\nabla F,\quad\nabla\cdot u=0\quad\text{in }\mathbb{R}^n,\\ & \lim u=0\text{ as }|x|\to\infty.\end{aligned} It is proved that for smooth bounded $$F\in L^1(\mathbb{R}^n)$$ the solution $$u$$ satisfying the conditions $u\in L^n(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n),\quad\nabla u\in L^{\frac n2}(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n),$ (the existence of such a solution has been established in the paper quoted above) satisfies the conditions $u\in L^{\frac n{n-1}}_w(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n),\quad\nabla u\in L^1_w(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n).$ $$L^p_w(\mathbb{R}^n)$$, $$0<p<\infty$$ denotes the space of measurable functions $$f$$ on $$\mathbb{R}^n$$ with finite functional $\sup_{t>0}\left[t\cdot \text{mes}\{x:|f(x)|>t\}^{\frac 1p}\right].$ The main part of the paper is concerned with the perturbation problem \begin{aligned}\frac{\partial u}{\partial t}+w\cdot\nabla u+u\cdot\nabla w+u\cdot\nabla u-\Delta u +\nabla p&=0,\quad x\in \mathbb{R}^n,\;t>0\\ \nabla\cdot u&=0,\quad x\in \mathbb{R}^n,\;t\geq 0\\ \left. u\right|_{t=0}=a,\quad\lim u=0\text{ as }|x|\to &\infty.\end{aligned} It is proved that if $$\sup|x||w(x)|+\sup|x|^2|\nabla w(x)|$$ is small then the weak solution $$u(t)\in H^1_\sigma$$ for all $$t\geq 0$$ and $$\lim\|u(t)\|_{H^1}=0$$ as $$t\to 0$$ ($$H^1$$ is the Hardy space). The analogous result is obtained for the space $$L^1$$. Explicit decay rates of the form $$\|u(t)\|=O(t^{-\frac\beta 2})$$, $$0<\beta\leq 1$$ are deduced for the solution under additional assumptions on $$w$$ and on the initial data $$a$$.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35B35 Stability in context of PDEs 76E99 Hydrodynamic stability
##### Keywords:
Navier-Stokes equations; Hardy spaces; stability; decay