Asymptotic Properties of Steady Plane Solutions of the Navier–Stokes Equations in a Cone-Like Domain

Motivated by Gilbarg–Weinberger’s early work on asymptotic properties of steady plane solutions of the Navier–Stokes equations on a neighborhood of infinity (Gilbarg andWeinberger in Ann Scuola Norm Super Pisa Cl Sci 5(2):381–404, 1978), we investigate asymptotic properties of steady plane solutions of this system on any cone-like domain of Ω0={(r,θ);r>r0,θ∈(0,θ0)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _0=\{(r,\theta ); r>r_0, \theta \in (0,\theta _0)\} $$\end{document} with finite Dirichlet integral and Navier-slip boundary conditions. It is proved that the velocity of the solution grows more slowly than logr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\log r}$$\end{document} as in Gilbarg andWeinberger (Ann Scuola Norm Super Pisa Cl Sci 5(2):381–404, 1978), while the mean value of the velocity converges to zero except the case of θ0=π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0=\pi $$\end{document}. Noting that Cauchy integral formula representation does not work in these domains due to the boundary obstacle, we explore some new technical lemmas to deal with these general cases. Moreover, Liouville type theorem on these domains and the decay estimates of the pressure or the vorticity are also obtained.