ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS

doi:10.1007/s10473-025-0210-y

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 446-472.doi: 10.1007/s10473-025-0210-y

ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS

Qunyi Bie¹, Hao Chen^2,*

1. College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. College of Science, China Three Gorges University, Yichang 443002, China

收稿日期:2022-11-20 修回日期:2023-01-08 出版日期:2025-03-25 发布日期:2025-05-08

ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS

Qunyi Bie¹, Hao Chen^2,*

1. College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. College of Science, China Three Gorges University, Yichang 443002, China

Received:2022-11-20 Revised:2023-01-08 Online:2025-03-25 Published:2025-05-08
Contact: *Hao Chen, E-mail: chenhao215@outlook.com
About author:Qunyi Bie, E-mail: qybie@126.com
Supported by:
The first author's work was supported by the National Natural Science Foundation of China (11871305).

摘要/Abstract

摘要： In this paper, we study the Cauchy problem of three-dimensional incompressible magnetohydrodynamics with almost symmetrical initial values in the cylindrical coordinates. Here the almost axisymmetric means that $(\partial_\theta u^r_0,\partial_\theta u^\theta_0,\partial_\theta u^z_0)$ is small. With additional smallness assumption on $(u^\theta_0,b^\theta_0)$, we prove the global existence of a unique strong solution $(\boldsymbol{u},\boldsymbol{b})$, which keeps close to some axisymmetric vector fields. Moreover, we give the initial data with some special symmetric structures that will persist for all time.

关键词: magnetohydrodynamics, almost symmetrical, cylindrical coordinates

Abstract: In this paper, we study the Cauchy problem of three-dimensional incompressible magnetohydrodynamics with almost symmetrical initial values in the cylindrical coordinates. Here the almost axisymmetric means that $(\partial_\theta u^r_0,\partial_\theta u^\theta_0,\partial_\theta u^z_0)$ is small. With additional smallness assumption on $(u^\theta_0,b^\theta_0)$, we prove the global existence of a unique strong solution $(\boldsymbol{u},\boldsymbol{b})$, which keeps close to some axisymmetric vector fields. Moreover, we give the initial data with some special symmetric structures that will persist for all time.

Key words: magnetohydrodynamics, almost symmetrical, cylindrical coordinates

中图分类号:

35Q35

Qunyi Bie, Hao Chen. ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS[J]. 数学物理学报(英文版), 2025, 45(2): 446-472.

Qunyi Bie, Hao Chen. ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS[J]. Acta mathematica scientia,Series B, 2025, 45(2): 446-472.

参考文献

[1] Ai X, Li Z.Global smooth solutions to the 3D non-resistive MHD equations with low regularity axisymmetric data. Commun Math Sci, 2022, 20: 1979-1994
[2] Chemin J Y, Gallagher I. Wellposedness and stability results for the Navier-Stokes equations in $\mathbb{R}^3$. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26(2): 599-624
[3] Chemin J Y, Gallagher I, Paicu M. Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann Math, 2011, 173: 983-1012
[4] Chemin J Y, McCormick D, Robinson J, Rodrigo J. Local existence for the non-resistive MHD equations in Besov spaces. Adv Math, 2016, 286: 1-31
[5] Danchin R, Tan J. On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces. Commun Partial Differ Equ, 2021, 46: 31-65
[6] Duvaut G, Lion J L. Inéquations en thermoélasticité et magnéohydrodynamique. Arch Ration Mech Anal, 1972, 46: 241-279
[7] Grimm R C, Dewar R L, Manickam J. Ideal MHD stability calculations in axisymmetric toroidal coordinate systems. J Comput, 1983, 49(1): 94-117
[8] Hasler U, Schneebeli A, Sch$\ddot{\mathrm{o}}$tzau D. Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl Numer Math, 2004, 51(1): 19-45
[9] He L B, Xu L, Yu P. On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves. Ann PDE, 2018, 4: 1-105[] Holst M, Lunasin E, Tsogtgerel G. Analysis of a general family of regularized Navier-Stokes and MHD models. J Nonlinear Sci, 2010, 20(5): 523-567
[10] Jiang Z, Zhu M. Regularity criteria for the 3D generalized MHD and Hall-MHD systems. Bull Malays Math Sci Soc, 2018, 41: 105-122
[11] Jiu Q, Liu J. Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Discrete Contin Dyn Syst, 2015, 35(1): 301-322
[12] Jiu Q, Yu H, Zheng X. Global well-posedness for axisymmetric MHD system with only vertical viscosity. J Differ Equ, 2017, 263(5): 2954-2990
[13] Ladyzhenskaya O A. On the unique global solvability to the Cauchy problem for the Navier-Stokes equations in the presence of the axial symmetry. Zap Nauchn Sem Leningrad Otdel Mat Inst Steklov, 1968, 7: 155-177
[14] Li Z, Liu W. Regularity criteria for the 3D axisymmetric non-resistive MHD system in Lorentz spaces. Results Math, 2023, 78(3): Art 86
[15] Lei Z. On axially symmetric incompressible magnetohydrodynamics in three dimensions. J Differ Equ, 2015, 259(7): 3202-3215
[16] Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math, 1934, 63: 193-248
[17] Liu Y. Global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field. Acta Appl Math, 2018, 155: 21-39
[18] Liu Y, Xu L. On the Existence and structures of almost axisymmetric solutions to 3-D Navier-Stokes equations. SIAM J Math Anal, 2023, 55(1): 458-485[] Sakakibara S, Watanabe K Y, Ohdachi S, et al. Study of MHD stability in LHD. Fusion Sci Technol, 2010, 58(1): 176-185
[19] Bedrossian J, Vicol V.The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction. Providence, RI: Amer Math Soc, 2022
[20] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36(5): 635-664
[21] Ukhovskii M R, Iudovich V I. Axially symmetric flows of ideal and viscous fluids filling the whole space. J Appl Math Mech, 1968, 32: 59-69
[22] Wang S, Wu J. Axially symmetric incompressible MHD in three dimensions. J Math Anal Appl, 2015, 426(1): 440-465
[23] Wang P, Guo Z. Global axisymmetric solutions to the 3D MHD equations with nonzero swirl. J Geom Anal, 2022, 32(10): Art 258
[24] Zhang Z, Yao Z. 3D axisymmetric MHD system with regularity in the swirl component of the vorticity. Comput Math Appl, 2017, 73(12): 2573-2580

Metrics

Viewed

Full text

Abstract

Just accepted	Online first	Issue

0	0	37

Cited

Shared

Discussed

ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS

ON ALMOST AXISYMMETRIC INCOMPRESSIBLE MAGNETOHYDRODYNAMICS IN THREE DIMENSIONS

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

Metrics

本文评价

推荐阅读 0

[1]	Zefu Feng, Jing Jia. THE GLOBAL WELL-POSEDNESS OF SOLUTIONS TO COMPRESSIBLE ISENTROPIC TWO-FLUID MAGNETOHYDRODYNAMICS IN A STRIP DOMAIN^*[J]. 数学物理学报(英文版), 2024, 44(5): 1997-2018.
[2]	Jae-Myoung KIM. LOCAL EXISTENCE AND BLOW-UP CRITERION OF 3D IDEAL MAGNETOHYDRODYNAMICS EQUATIONS[J]. 数学物理学报(英文版), 2018, 38(6): 1759-1766.
[3]	秦绪龙, 杨彤, 姚正安, 周文书. A STUDY ON THE BOUNDARY LAYER FOR THE PLANAR MAGNETOHYDRODYNAMICS SYSTEM[J]. 数学物理学报(英文版), 2015, 35(4): 787-806.
[4]	高真圣, 谭忠, 吴国春. GLOBAL EXISTENCE AND CONVERGENCE RATES OF SMOOTH SOLUTIONS FOR THE 3-D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITHOUT HEAT CONDUCTIVITY[J]. 数学物理学报(英文版), 2014, 34(1): 93-106.
[5]	徐新英. A BLOW-UP CRITERION FOR 3-D NON-RESISTIVE COMPRESSIBLE HEAT-CONDUCTIVE MAGNETOHYDRODYNAMIC EQUATIONS#br# WITH INITIAL VACUUM[J]. 数学物理学报(英文版), 2012, 32(5): 1883-1900.
[6]	Christian Klingenberg, Knut Waagan. RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS -A REVIEW[J]. 数学物理学报(英文版), 2010, 30(2): 621-632.
[7]	Blake Barker, Olivier Lafitte, Kevin Zumbrun. EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY[J]. 数学物理学报(英文版), 2010, 30(2): 447-498.
[8]	Philippe G. LeFloch, Siddhartha Mishra. KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT[J]. 数学物理学报(英文版), 2009, 29(6): 1684-1702.