四维定常 Navier-Stokes 方程离散自相似解的存在性——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 709-723.

四维定常 Navier-Stokes 方程离散自相似解的存在性——献给陈化教授 70 寿辰

刘浩¹(), 王云²(), 谢春景³^,^*()

¹ 澳门大学理工学院数学系澳门 999078
² 苏州大学数学科学学院、动力系统与微分方程研究中心苏州 215006
³ 上海交通大学数学科学学院, 自然科学研究院, 教育部科学工程计算重点实验室, 上海市现代分析前沿研究基地上海 200240

收稿日期:2025-12-31 修回日期:2026-02-24 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 谢春景 E-mail:haoliu@um.edu.mo;ywang3@suda.edu.cn;cjxie@sjtu.edu.cn
作者简介:刘浩, Email：haoliu@um.edu.mo
王云, Email：ywang3@suda.edu.cn
基金资助:
国家自然科学基金(12271389);国家自然科学基金(12250710674);国家自然科学基金(12571238);国家自然科学基金(12426203);和江苏省自然科学基金(BK20240147)

Existence of Discretely Self-Similar Solutions to Four-Dimensional Steady Navier-Stokes Equations

Hao Liu¹(), Yun Wang²(), Chunjing Xie³^,^*()

¹ Department of Mathematics, University of Macau, Taipa, Macau 999078
² School of Mathematical Sciences, Center for Dynamical System and Differential Equations, Soochow University, Suzhou 215006
³ School of Mathematical Sciences, Institute of Natural Sciences, Key Laboratory of Scientific and Engineering Computing of the Ministry of Education, Shanghai Frontiers Research Center for Modern Analysis, Shanghai Jiao Tong University, Shanghai 200240

Received:2025-12-31 Revised:2026-02-24 Online:2026-04-26 Published:2026-04-27
Contact: Chunjing Xie E-mail:haoliu@um.edu.mo;ywang3@suda.edu.cn;cjxie@sjtu.edu.cn
Supported by:
NSFC(12271389);NSFC(12250710674);NSFC(12571238);NSFC(12426203);Natural Science Foundation of Jiangsu Province(BK20240147)

摘要/Abstract

摘要：

该文证明了对于任意给定的局部 Lipschitz 的离散自相似外力, 四维空间上定常 Navier-Stokes 方程至少存在一个离散自相似解. 若外力在除掉原点以外光滑, 作者构造的离散自相似解在除掉原点以外也光滑. 值得一提的是, 这里解的存在性结果不需要对外力作任何小性假设.

关键词: 定常 Navier-Stokes 方程, 离散自相似解, 四维, 存在性

Abstract:

In this paper, we prove that there exists at least one discretely self-similar solution to the steady Navier-Stokes equations in $\mathbb{R}^4\backslash\{0\}$ for any given locally Lipschitz discretely self-similar external force. If the external force is smooth away from the origin, the constructed discretely self-similar solution is also smooth away from the origin. Notably, the existence result does not require any smallness assumption on the external force.

Key words: steady Navier-Stokes equations, discretely self-similar solutions, four dimension, existence

中图分类号:

O175.29

刘浩, 王云, 谢春景. 四维定常 Navier-Stokes 方程离散自相似解的存在性——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 709-723.

Hao Liu, Yun Wang, Chunjing Xie. Existence of Discretely Self-Similar Solutions to Four-Dimensional Steady Navier-Stokes Equations[J]. Acta mathematica scientia,Series A, 2026, 46(2): 709-723.

参考文献 31

[1]	Bang J, Gui C, Liu H, et al. On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions. arXiv:2510.10488
[2]	Bang J, Gui C, Liu H, et al. Rigidity of steady solutions to the Navier-Stokes equations in high dimensions and its applications. J Eur Math Soc, 2025. DOI:10.4171/JEMS/1738
[3]	Frehse J, R{\r u}ži\v{c}ka M. Regularity for the stationary Navier-Stokes equations in bounded domains. Arch Ration Mech Anal, 1994, 128(4): 361-380
[4]	Frehse J, R{\r u}ži\v{c}ka M. Existence of regular solutions to the steady Navier-Stokes equations in bounded six-dimensional domains. Ann Scuola Norm Sup Pisa Cl Sci (4), 1996, 23(4): 701-719
[5]	Frehse J, R{\r u}{ž}i{\v{c}}ka M. On the regularity of the stationary Navier-Stokes equations. Ann Scuola Norm Sup Pisa Cl Sci (4), 1994, 21(1): 63-95
[6]	Frehse J, R{\r u}{ž}i{\v{c}}ka M. A new regularity criterion for steady Navier-Stokes equations. Differential Integral Equations, 1998, 11(2): 361-368
[7]	Galdi G P. On the Oseen boundary value problem in exterior domains//The Navier-Stokes Equations II——Theory and Numerical Methods. Berlin: Springer, 1992
[8]	Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer Monographs in Mathematics, New York: Springer, 2011
[9]	Gerhardt C. Stationary solutions to the Navier-Stokes equations in dimension four. Math Z, 1979, 165(2): 193-197
[10]	Glazier J A, Libchaber A. Quasi-periodicity and dynamical systems: an experimentalist's view. IEEE Trans Circuits and Systems, 1988, 35(7): 790-809
[11]	Guillod J, Wittwer P. Generalized scale-invariant solutions to the two-dimensional stationary Navier-Stokes equations. SIAM J Math Anal, 2015, 47(1): 955-968
[12]	Hamel G. Spiralförmige bewegungen zäher flüssigkeiten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1917, 25: 34-60
[13]	Jeffery G B. The two-dimensional steady motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1915, 29: 455-465
[14]	Kaneko K, Kozono H, Shimizu S. Stationary solution to the Navier-Stokes equations in the scaling invariant Besov space and its regularity. Indiana Univ Math J, 2019, 68(3): 857-880
[15]	Korolev A, Šverák V. On the large-distance asymptotics of steady state solutions of the Navier-Stokes equations in 3D exterior domains. Ann Inst H Poincaré C Anal Non Linéaire, 2011, 28(2): 303-313
[16]	Kozono H, Yamazaki M. The stability of small stationary solutions in Morrey spaces of theNavier-Stokes equation. Indiana Univ Math J, 1995, 44(4): 1307-1336
[17]	Landau L. A new exact solution of Navier-Stokes equations. C R (Doklady) Acad Sci URSS (NS), 1944, 43: 286-288
[18]	Leray J. étude de diverses équations intégrales non linéaireset de quelques problèmes que pose l'hydrodynamique. J Math Pures Appl, 1933, 12: 1-82
[19]	Li L, Li Y, Yan X. Homogeneous solutions of stationary Navier-Stokes equations withisolated singularities on the unit sphere. I. One singularity. Arch Ration Mech Anal, 2018, 227(3): 1091-1163
[20]	Li Y, Yang Z. Regular solutions of the stationary Navier-Stokes equations onhigh dimensional Euclidean space. Comm Math Phys, 2022, 394(2): 711-734
[21]	Miura H, Tsai T P. Point singularities of 3D stationary Navier-Stokes flows. J Math Fluid Mech, 2012, 14(1): 33-41
[22]	Phan T V, Phuc N C. Stationary Navier-Stokes equations with critically singularexternal forces: Existence and stability results. Adv Math, 2013, 241: 137-161
[23]	Shi Z. Self-similar solutions of stationary Navier-Stokes equations. J Differential Equations, 2018, 264(3): 1550-1580
[24]	Sornette D. Discrete-scale invariance and complex dimensions. Phys Rep, 1998, 297(5): 239-270
[25]	Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press Princeton NJ, 1993
[26]	Struwe M. Regular solutions of the stationary Navier-Stokes equations on $\mathbb{R}^5$. Math Ann, 1995, 302(4): 719-741
[27]	Šverák V. On Landau's solutions of the Navier-Stokes equations. J Math Sci, 2011, 179(1): 208-228
[28]	Tian G, Xin Z. One-point singular solutions to the Navier-Stokes equations. Topol Methods Nonlinear Anal, 1998, 11(1): 135-145
[29]	Tsai T P. On Problems Arising in the Regularity Theory for theNavier-Stokes Equations. Ann Arbor: University of Minnesota, 1998
[30]	Tsai T P. Lectures on Navier-Stokes Equations. Providence, RI: American Mathematical Society, 2018
[31]	Tsurumi H. Well-posedness and ill-posedness problems of the stationaryNavier-Stokes equations in scaling invariant Besov spaces. Arch Ration Mech Anal, 2019, 234(2): 911-923