New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations

Abstract

The Liouville problem for the three-dimensional stationary Navier--Stokes equations remains open, even for axisymmetric \(D\)-solutions. In this paper, we obtain two results based on decay in the cylindrical radial variable \(r=|x'|\). (i). Using a new pointwise Calder\'on--Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo--Pan--Zhang (2020, JFA) and prove \[ |\nabla u_r|+|\nabla u_z| \lesssim r^{-5/4}[\log(\mathrm e+r)]^{5/4}, \quad |\omega_r|+|\omega_z| \lesssim r^{-9/8}[\log(\mathrm e+r)]^{9/8}, \quad r\gg1. \] (ii). We develop a new approach to Liouville theorems that improves the axisymmetric criteria of Wang (2019, JDE) and Zhao (2019, Nonlinear Anal.). Without any symmetry assumption, we show that a \(D\)-solution is trivial if one of the following holds: \[ (\mathrm a).\,\sup_{{|x'|=r,\, z\in\mathbb R}} |u(x',z)| \leq Cr^{-2/3}[\log(\mathrm e+r)]^{-\gamma}; \quad (\mathrm b).\, \sup_{{|x'|=r,\, z\in\mathbb R}} |\omega(x',z)| \leq Cr^{-5/3}[\log(\mathrm e+r)]^{-\gamma}, \] for $r\geq1$, where $\gamma>1/3$.