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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 \(a\).

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B35 Stability in context of PDEs
76E99 Hydrodynamic stability