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

2020/01/02 by Yueyang Men, Men, Yueyang, Wendong Wang +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2001.00377

openalex publication_date 2020/01/02 · openalex created_date 2020/01/10 · openalex updated_date 2026/07/28

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⋅∇ u=0 decays in the form of exp(-C|x|log²|x|) at infinity provided ‖W‖_{L^∞(ℝ²∖ B1)}\lesssim 1, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by Kenig-Wang \citeKW2015 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 \citeKL2019 to exp(-C|x|log²|x|), where the minimal decaying rate of exp(-C|x|^\frac32+) is obtained by Kow-Lin in a recent paper \citeKL2019 by using appropriate Carleman estimates.

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