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 $-\triangle u+W\cdot\nabla u=0$ decays in the form of $\exp(-C|x|\log^2|x|)$ at infinity provided $\|W\|_{L^\infty(\mathbb{R}^2\setminus B_1)}\lesssim 1$, 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 $\exp(-C|x|^{\frac32+})$ is obtained by Kow-Lin in a recent paper \cite{KL2019} by using appropriate Carleman estimates.