Liouville type theorems on the steady Navier-Stokes equations in R3
Liouville type theorems on the steady Navier-Stokes equations in R3
复制标题
R3 中稳态纳维-斯托克斯方程的刘维尔型定理
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Deliang Xu
中科院分区：
文献类型：
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作者：
Z. Xin;Deliang Xu
本文研究了$\mathbf{R}^{3}$中定常不可压Navier-Stokes方程解的Liouville型性质。证明了在$\mathbf{R}^{3}$中定常Navier-Stokes方程的任意解都是平凡的，其中Dirichlet积分有限且远场速度场为零.这解决了一个开放的问题。证明的关键成分包括能量通量的霍奇分解和变形矩阵的平方位于局部哈代空间的观察。作为副产品，我们还获得了一个Liouville型定理的定常密度相关的Navier-Stokes方程。
In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in $\mathbf{R}^{3}$. It is shown that any solution to the steady Navier-Stokes equations in $\mathbf{R}^{3}$ with finite Dirichlet integral and vanishing velocity field at far fields must be trivial. This solves an open problem. The key ingredients of the proof include a Hodge decomposition of the energy-flux and the observation that the square of the deformation matrix lies in the local Hardy space. As a by-product, we also obtain a Liouville type theorem for the steady density-dependent Navier-Stokes equations.