Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain

Abstract

We consider an elliptic equation with unbounded drift in an exterior domain, and obtain quantitative uniqueness estimates at infinity, i.e. the non-trivial solution of $-△u+W\cdot \nabla u=0$ decays in the form of $\exp (-C|x|{\log }^2|x|)$ at infinity provided $\Vert W{\Vert }_{L^{\infty }({\mathbb{R}}^2\setminus B_1)}\lesssim 1$, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by Kenig-Wang \cite{KW2015} in the whole space. As an application, the asymptotic behavior of an incompressible fluid around a bounded obstacle is also considered. Specially for the two-dimensional case, we can improve the decay rate in \cite{KL2019} to $\exp (-C|x|{\log }^2|x|)$, where the minimal decaying rate of $\exp (-C|x{|}^{\frac{3}{2}+})$ is obtained by Kow-Lin in a recent paper \cite{KL2019} by using appropriate Carleman estimates.