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May  2021, 41(5): 2301-2340. doi: 10.3934/dcds.2020366

## On the asymptotic properties for stationary solutions to the Navier-Stokes equations

 Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA

* Corresponding author: Oleg Imanuvilov

Received  August 2019 Revised  July 2020 Published  May 2021 Early access  November 2020

Fund Project: The author is supported by NSF grant DMS 1312900

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 $v$ and its derivatives decay like $O(|x|^{-k})$ for sufficiently large $k$, depending on the velocity field, as $|x|\to \infty$, then $v$ is zero on that exterior domain. Constructive estimate for $k$ is given. In the case where velocity field is only bounded at infinity, we show that the infimum of $L^2$ norm of a velocity field on a unit ball located at distance $t$ from an origin is bounded from below as $Ce^{-\beta t^\frac 43\ln(t)}.$ The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.

Citation: Oleg Imanuvilov. On the asymptotic properties for stationary solutions to the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2301-2340. doi: 10.3934/dcds.2020366
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