三维轴对称定常Navier-Stokes方程的新衰减估计与Liouville型定理
New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations
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中文总结 AI 辅助
本文针对三维轴对称定常Navier-Stokes方程，改进了衰减估计与Liouville型定理，放宽轴对称准则，在无对称假设下得到D-解平凡的判定条件。
中文摘要 AI 辅助
三维定常Navier-Stokes方程的Liouville问题仍未解决，即使是轴对称D-解的情况也不例外。本文基于柱径向变量r=|x'|的衰减性得到两个结果：(i) 利用适配柱几何的新型点态Calderón-Zygmund估计，改进了Carrillo-Pan-Zhang（2020，JFA）的衰减估计，证明当r≫1时，|∇uᵣ|+|∇u_z|≲r⁻⁵/⁴[log(e+r)]⁵/⁴，|ωᵣ|+|ω_z|≲r⁻⁹/⁸[log(e+r)]⁹/⁸；(ii) 开发了一种新的Liouville定理方法，改进了Wang（2019，JDE）和Zhao（2019，Nonlinear Anal.）的轴对称准则，在无任何对称性假设下，当r≥1且γ>1/3时，若满足以下任一条件：(a) sup_{|x'|=r,z∈ℝ}|u(x',z)|≤Cr⁻²/³[log(e+r)]⁻γ；(b) sup_{|x'|=r,z∈ℝ}|ω(x',z)|≤Cr⁻⁵/³[log(e+r)]⁻γ，则D-解是平凡的。
英文摘要
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 $γ>1/3$.
