THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

Abstract

References

Related Articles 15

Metrics

Comments

Recommended 10

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

doi:10.1007/s10473-024-0311-z

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (3): 973-983.doi: 10.1007/s10473-024-0311-z

Previous Articles Next Articles

Cuntao xiao¹, Hua qiu^2,*, Zheng-an yao³

1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510090, China;
2. Department of Mathematics, South China Agricultural University, Guangzhou 510642, China;
3. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received:2022-07-08 Revised:2022-11-25 Online:2024-06-25 Published:2024-05-21
Contact: *Hua qiu, E-mail:tsiuhua@scau.edu.cn
About author:Cuntao xiao,E-mail:xiaocuntao@gdut.edu.cn; Zheng-an yao ,E-mail:mcsyao@mail.sysu.edu.cn
Supported by:
Xiao's work was supported by the Opening Project of Guangdong Province Key Laboratory of Cyber-Physical System (2016B030301008); Qiu's work was supported by the National Natural Science Foundation of China (11126266), the Natural Science Foundation of Guangdong Province (2016A030313390), the Quality Engineering Project of Guangdong Province (SCAU-2021-69), and the SCAU Fund for High-level University Building; Yao's work was supported by the National Key Research and Development Program of China(2020YFA0712500) and the National Natural Science Foundation of China (11971496, 12126609).

Abstract: In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

Key words: regularized MHD equations, fractional Laplacian, global well-posedness, analyticity, decay rate

CLC Number:

35B65

Cuntao xiao, Hua qiu, Zheng-an yao. THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS[J].Acta mathematica scientia,Series B, 2024, 44(3): 973-983.

Trendmd

[1] Bae H. Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations. Proc Amer Math Soc, 2015, 143(7): 2887-2892
[2] Benameur J. Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations. J Math Anal Appl, 2015, 422: 424-434
[3] Benameur J, Bennaceur M. Large time behaviour of solutions to the 3D-NSE in

$\chi^{\delta}$ spaces. J Math Anal Appl, 2020, 482: 123566
[4] Catania D. Finite dimensional global attractor for 3D MHD-

$\alpha$ models: a comparison. J Math Fluid Mech, 2012, 14: 95-115
[5] Chen S Y, Foias C, Holm D D, et al. Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys Rev Lett, 1998, 81(24): 5338-5341
[6] Du Y, Qiu H, Yao Z. Global well-posedness of the Cauchy problem for certain magnetohydrodynamic-

$\alpha$ models. Math Meth Appl Sci, 2010, 33(13): 1545-1557
[7] Fan J, Ozawa T. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-

$\alpha$ -MHD model. Kinetic Related Models, 2009, 2: 293-305
[8] Fan J, Ozawa T. Global Cauchy problem for the 2-D magnetohydrodynamic-

$\alpha$ models with partial viscous terms. J Math Fluid Mech, 2010, 12: 306-319
[9] Holm D D. Average Lagrangians and the mean effects of fluctuations in ideal fluid dynamics. Physica D, 2002, 170: 253-286
[10] Jasmine S L, Titi E S. Analytical study of certain magnetodrodynamic-

$\alpha$ models. J Math Phy, 2007, 48: 065504
[11] Lei Z, Lin F. Global mild solutions of Navier-Stokes equations. Comm Pure Appl Math, 2011, 64: 1297-1304
[12] Wang Y. Asymptotic decay of solutions to 3D MHD equations. Nonliear Anal: TMA, 2016, 132: 115-125
[13] Wang Y, Wang K. Global well-posedness of the three dimensional magnetohydrodynamics equations. Nonlinear Anal: RWA, 2014, 17: 245-251
[14] Xiao Y, Yuan B, Zhang Q. Temporal decay estimate of solutions to 3D generalized magnetohydrodynamic system. Appl Math Lett, 2019, 98: 108-113
[15] Xu X, Ye Z. Note on global regularity of 3D generalized magnetohydrodynamic-

$\alpha$ model with zero diffusivity. Comm Pure Appl Anal, 2015, 14: 585-595
[16] Yamazaki K. A remark on the two-dimensional magnetohydrodynamics-alpha system. J Math Fluid Mech, 2016, 18(3): 609-623
[17] Ye Z. Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations. Ann Mat Pura Appl, 2016, 195: 1111-1121
[18] Ye Z, Xu X. Global regularity of 3D generalized incompressible magnetohydrodynamic-

$\alpha$ model. Appl Math Lett, 2014, 35: 1-6
[19] Ye Z, Zhao X. Global well-posedness of the generalized magnetohydrodynamic equations. Z Angew Math Phys, 2018, 69: Art 126
[20] Yu Y, Li K. Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations. J Math Anal Appl, 2007, 329: 298-326
[21] Zhao J, Zhu M. Global regularity for the incompressible MHD-

$\alpha$ system with fractional diffusion. Appl Math Lett, 2014, 29: 26-29
[22] Zhao X, Cao J. Global well-posedness of solutions for magnetohydrodynamics-

$\alpha$ model in

$\mathbb{R}^3$ . Appl Math Lett2017, 74: 134-139
[23] Zhou Y, Fan J.Global well-posedness for two modified-Leray-

$\alpha$ -MHD models with partial viscous terms. Math Methods Appl Sci, 33(7): 856-862
[24] Zhou Y, Fan J. On the Cauchy problem for a Leray-

$\alpha$ -MHD model. Nonlinear Anal: RWA, 2011, 12: 648-657

[1]	Ying Wang, Yanjing Qiu, Qingping Yin. THE RADIAL SYMMETRY OF POSITIVE SOLUTIONS FOR SEMILINEAR PROBLEMS INVOLVING WEIGHTED FRACTIONAL LAPLACIANS [J]. Acta mathematica scientia,Series B, 2024, 44(3): 1020-1035.
[2]	Han Wang, Yinghui Zhang. THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING [J]. Acta mathematica scientia,Series B, 2024, 44(3): 1064-1095.
[3]	Zhenhua Guo, Xueyao Zhang. INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^* [J]. Acta mathematica scientia,Series B, 2024, 44(1): 247-274.
[4]	Meiqing XU. ON A SUPER POLYHARMONIC PROPERTY OF A HIGHER-ORDER FRACTIONAL LAPLACIAN^* [J]. Acta mathematica scientia,Series B, 2023, 43(6): 2589-2596.
[5]	Mingjuan Chen, Shuai Zhang. ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA^* [J]. Acta mathematica scientia,Series B, 2023, 43(5): 2215-2233.
[6]	Léo Glangetas, Van-Sang Ngo, El Mehdi Said. HYDROSTATIC LIMIT OF THE NAVIER-STOKES-ALPHA MODEL^* [J]. Acta mathematica scientia,Series B, 2023, 43(5): 1945-1980.
[7]	Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao. GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN ${\mathbb{R}}^{3}$ ^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1301-1322.
[8]	Huafei DI, Weijie Rong. THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE* [J]. Acta mathematica scientia,Series B, 2023, 43(1): 324-348.
[9]	Weixi LI, Rui XU, Tong YANG. GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES [J]. Acta mathematica scientia,Series B, 2022, 42(6): 2343-2366.
[10]	Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL [J]. Acta mathematica scientia,Series B, 2022, 42(6): 2230-2256.
[11]	Chao Jiang, Zuhan Liu, Ling Zhou. BLOW-UP IN A FRACTIONAL LAPLACIAN MUTUALISTIC MODEL WITH NEUMANN BOUNDARY CONDITIONS [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1809-1816.
[12]	Qingyou He, Jiawei Sun. LARGE TIME BEHAVIOR OF THE 1D ISENTROPIC NAVIER-STOKES-POISSON SYSTEM [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1843-1874.
[13]	Wenjun WANG, Zhen CHENG. LARGE TIME BEHAVIOR OF GLOBAL STRONG SOLUTIONS TO A TWO-PHASE MODEL WITH A MAGNETIC FIELD [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1921-1946.
[14]	Jinlu LI, Zhaoyang YIN, Xiaoping ZHAI. GLOBAL WELL-POSEDNESS FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS [J]. Acta mathematica scientia,Series B, 2022, 42(5): 2131-2148.
[15]	Xueting Jin, Yuelong Xiao, Huan Yu. GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION [J]. Acta mathematica scientia,Series B, 2022, 42(4): 1293-1309.

Viewed

Full text

	From	local

	Times	1
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	52

	From	Others

	Times	52
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	Ang FU, Mingjin LI, Di YANG. FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY[J]. Acta mathematica scientia,Series B, 2024, 44(2): 405 -419 .
[2]	Changlin XIANG, Gaofeng ZHENG. SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION[J]. Acta mathematica scientia,Series B, 2024, 44(2): 420 -430 .
[3]	Zihou ZHANG, Jing ZHOU. THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS[J]. Acta mathematica scientia,Series B, 2024, 44(2): 445 -454 .
[4]	Yinzheng Sun, Aifang Qu, Hairong Yuan. THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 37 -77 .
[5]	Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL[J]. Acta mathematica scientia,Series B, 2024, 44(2): 455 -473 .
[6]	Huifang Liu, Zhiqiang Mao. THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 103 -114 .
[7]	Zhong Tan, Xinliang Li, Hui Yang. ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW[J]. Acta mathematica scientia,Series B, 2024, 44(3): 851 -864 .
[8]	Bin Han, Ningan Lai, Andrei Tarfulea. THE GLOBAL EXISTENCE OF STRONG SOLUTIONS FOR A NON-ISOTHERMAL IDEAL GAS SYSTEM[J]. Acta mathematica scientia,Series B, 2024, 44(3): 865 -886 .
[9]	Weiyuan Zou. THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THERMOMECHANICAL CUCKER-SMALE-STOKES QUATIONS IN THE WHOLE DOMAIN[J]. Acta mathematica scientia,Series B, 2024, 44(3): 887 -908 .
[10]	Xiaoshan Wang, Zhongqian Wang, Zhe Jia. GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH $p$ -LAPLACIAN DIFFUSION AND LOGISTIC SOURCE[J]. Acta mathematica scientia,Series B, 2024, 44(3): 909 -924 .

Just accepted

Online first

Just accepted

Online first