| Popis: |
In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution \begin{document}$ v $\end{document} and its derivatives decay like \begin{document}$ O(|x|^{-k}) $\end{document} for sufficiently large \begin{document}$ k $\end{document} , depending on the velocity field, as \begin{document}$ |x|\to \infty $\end{document} , then \begin{document}$ v $\end{document} is zero on that exterior domain. Constructive estimate for \begin{document}$ k $\end{document} is given. In the case where velocity field is only bounded at infinity, we show that the infimum of \begin{document}$ L^2 $\end{document} norm of a velocity field on a unit ball located at distance \begin{document}$ t $\end{document} from an origin is bounded from below as \begin{document}$ Ce^{-\beta t^\frac 43\ln(t)}. $\end{document} The proof of these results are based on the Carleman type estimates, and also the Kelvin transform. |
