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arXiv · 2608.06040

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

Wendong Wang·Guoxu Yang
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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 Calderón--Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo--Pan--Zhang (2020, JFA) and prove \[ |\nabla u_r|+|\nabla u_z| \lesssim r^{-5/4}[\log(\mathrm e+r)]^{5/4}, \quad |ω_r|+|ω_z| \lesssim r^{-9/8}[\log(\mathrm e+r)]^{9/8}, \quad r\gg1. \] (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: \[ (\mathrm a).\,\sup_{|x'|=r,\, z\in\mathbb R} |u(x',z)| \leq Cr^{-2/3}[\log(\mathrm e+r)]^{-γ}; \quad (\mathrm b).\, \sup_{|x'|=r,\, z\in\mathbb R} |ω(x',z)| \leq Cr^{-5/3}[\log(\mathrm e+r)]^{-γ}, \] for $r\geq1$, where $γ>1/3$.

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Wendong Wang, Guoxu Yang. 2026-08-06. New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations. https://arxiv.org/abs/2608.06040

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