One-point singular solutions to the Navier-Stokes equations
One-point singular solutions to the Navier-Stokes equations
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DOI：
10.12775/tmna.1998.008
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发表时间：
1998-03
影响因子：
0.7
通讯作者：
G. Tian;Z. Xin
中科院分区：
文献类型：
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作者：
G. Tian;Z. Xin
具有适当齐次性的平稳解或自相似解在非线性问题的正则性理论中起着至关重要的作用，这些问题在物理上或几何上都是有趣的。这已经在调和映射和极小曲面的正则性理论中得到了证明。[CKN]中的局部部分正则性定理意味着不存在具有小局部能量的自相似解（也参见[TX]的推广）。利用[NRS]中的一些论点，Tsai已经排除了任何具有有限局部能量的自相似解的存在性。然而，在三维空间中的不可压缩Navier-Stokes方程的解是否会在有限时间内发展出奇点，这一点还不清楚。因此，构造三维Navier-Stokes方程的特解可能仍然是有趣的。本文构造了三维不可压缩Navier-Stokes方程在R\p上的一个单参数显式光滑解族，其中p为任意给定点.这些解是轴对称的，齐次的-1次。它们是Navier-Stokes方程的定常解，也解决了自相似问题。
Stationary or self similar solutions with suitable homogeneity often play a crucial role in the regularity theory of nonlinear problems, which are physically or geometrically interesting. This has been manifested in the regularity theory of harmonic maps and minimal surfaces. The local partial regularity theorem in [CKN] implies that there are no self-similar solutions with small local energy (also see [TX] for generalizations). Making use of some arguments in [NRS], Tsai has ruled out the existence of any self-similar solutions with a finite local energy. Yet it is unclear whether or not solutions of the incompressible Navier– Stokes equation in three space dimensions would develop singularities in finite time. Therefore, it may be still interesting to construct special solutions of the 3-dimensional Navier–Stokes equation. In this note, we construct a one-parameter family of explicit smooth solutions of the 3-dimensional impressible Navier–Stokes equation on R\p, where p is any given point. These solutions are axisymmetric, homogeneous of degree −1. They are steady solutions to the Navier–Stokes equations and also solve the self-similar