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| We consider the following initial value problem for the Navier-Stokes system for an incompressible fluid in the whole three dimensional space
\begin{eqnarray*}
u_t-\Delta u+ \nabla \cdot (u \otimes u )+\nabla p&=&F,\quad (x,t) \in {\bf R}^3\times (0,\infty )\\
{\rm div }\ u&=&0, \\
u(x,0)&=&u_0 (x).
\end{eqnarray*}
It is well-known that this problem has a unique global-in-time mild solution for a sufficiently small initial condition $u_0$ and for a small external force $F$ in suitable scaling invariant spaces. We show that these global-in-time mild solutions are asymptotically stable under every (arbitrary large) $L^2$-perturbation of their initial conditions. |
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