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

TL;DR

This paper investigates the decay rates of solutions to an elliptic PDE with unbounded drift in exterior domains, establishing sharp quantitative decay estimates and extending previous results to the case of fluid flow around obstacles.

Contribution

It provides sharp decay estimates for solutions of elliptic equations with unbounded drift in exterior domains, generalizing prior decay theorems and improving decay rates for fluid flow around obstacles.

Findings

Solutions decay as exp(-C|x| log^2|x|) at infinity.

Decay estimates are sharp, supported by counterexamples.

Improved decay rate for 2D fluid flow around obstacles.

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.