WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES

doi:10.1007/s10473-019-0607-6

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (6): 1551-1567.doi: 10.1007/s10473-019-0607-6

WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES

Azzeddine EL BARAKA, Mohamed TOUMLILIN

FST FES, Laboratory AAFA, Department of Mathematics, University Sidi Mohamed Ben Abdellah, B. P 2202 Route Immouzer Fes 30000, Morocco

收稿日期:2018-07-31 修回日期:2019-05-09 出版日期:2019-12-25 发布日期:2019-12-30
通讯作者: Mohamed TOUMLILIN,E-mail:mohamed.toumlilin@usmba.ac.ma E-mail:mohamed.toumlilin@usmba.ac.ma
作者简介:Azzeddine EL BARAKA,E-mail:azzeddine.elbaraka@usmba.ac.ma

WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES

Azzeddine EL BARAKA, Mohamed TOUMLILIN

FST FES, Laboratory AAFA, Department of Mathematics, University Sidi Mohamed Ben Abdellah, B. P 2202 Route Immouzer Fes 30000, Morocco

Received:2018-07-31 Revised:2019-05-09 Online:2019-12-25 Published:2019-12-30
Contact: Mohamed TOUMLILIN,E-mail:mohamed.toumlilin@usmba.ac.ma E-mail:mohamed.toumlilin@usmba.ac.ma

摘要/Abstract

摘要： This paper concerns the Cauchy problem of the 3D generalized incompressible magneto-hydrodynamic (GMHD) equations. By using the Fourier localization argument and the Littlewood-Paley theory, we get local well-posedness results of the GMHD equations with large initial data (u₀, b₀) belonging to the critical Fourier-Besov-Morrey spaces FṄ_p,λ,q^{1-2α+ 3/p'+λ/p} (R³). Moreover, stability of global solutions is also discussed.

关键词: magneto-hydrodynamic, Fourier-Besov-Morrey space, stability

Abstract: This paper concerns the Cauchy problem of the 3D generalized incompressible magneto-hydrodynamic (GMHD) equations. By using the Fourier localization argument and the Littlewood-Paley theory, we get local well-posedness results of the GMHD equations with large initial data (u₀, b₀) belonging to the critical Fourier-Besov-Morrey spaces FṄ_p,λ,q^{1-2α+ 3/p'+λ/p} (R³). Moreover, stability of global solutions is also discussed.

Key words: magneto-hydrodynamic, Fourier-Besov-Morrey space, stability

中图分类号:

35Q35

Azzeddine EL BARAKA, Mohamed TOUMLILIN. WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES[J]. 数学物理学报(英文版), 2019, 39(6): 1551-1567.

Azzeddine EL BARAKA, Mohamed TOUMLILIN. WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES[J]. Acta mathematica scientia,Series B, 2019, 39(6): 1551-1567.

参考文献

[1] Bahouri H, Chemin J Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wis-senschaften, Vol 343. New York:Springer-Verlag, 2011
[2] Benameur J. Long time decay to the LeiLin solution of 3D Navier-Stokes equations. J Math Anal Appl, 2015, 422(1):424-434
[3] Bie Q, Wang Q, Yao Z A. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinet Relat Mod, 2015, 8(3):395-411
[4] Cannone M, Miao C, Prioux N, Yuan B. The Cauchy problem for the magneto-hydrodynamic system. Banach Center Publ, 2006, 74:59-93
[5] Cannone M, Wu G. Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces. Nonlinear Anal, 2012, 75:3754-3760
[6] Cao C, Regmi D, Wu J. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J Differ Equ, 2013, 254:2661-2681
[7] Cao C, Wu J. Two regularity criteria for the 3D MHD equations. J Differ Equ, 2010, 248:2263-2274
[8] Cao C, Wu J. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv Math, 2011, 226:1803-1822
[9] Cao C, Wu J, Yuan B. The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J Math Anal, 2014, 46(1):588-602
[10] Chen Q, Miao C, Zhang Z. On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamics equations. Commun Math Phys, 2008, 284:919-930
[11] El Baraka A, Toumlilin M. Global well-posedness for fractional Navier-Stokes equations in critical FourierBesov-Morrey spaces. Moroccan J Pure and Appl Anal, 2017, 3(1):1-14
[12] El Baraka A, Toumlilin M. Global well-posedness and decay results for 3D generalized magnetohydrodynamic equations in critical Fourier-Besov-Morrey spaces. Electron J Differ Equ, 2017, 2017(65):1-20
[13] Ferreira L C, Lima L S. Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces. Monatsh Math, 2014, 175(4):491-509
[14] Gallagher I, Iftimie D, Planchon F. Non-blowup at large times and stability for global solutions to the Navier-Stokes equations. CR Math Acad Sci Paris, 2002, 334(4):289-292
[15] He C, Wang Y. On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J Differ Equ, 2007, 238:1-17
[16] He C, Xin Z. On the regularity of solutions to the magnetohydrodynamic equations. J Differ Equ, 2005, 213:235-254
[17] Kato T. Strong solutions of the Navier-Stokes equations in Morrey spaces. Bol Soc Brasil Mat, 1992, 22(2):127-155
[18] Lei Z. On axially symmetric incompressible magnetohydrodynamics in three dimensions. J Differ Equ, 2015, 4259(7):3202-3215
[19] Lei Z, Lin F. Global mild solutions of Navier-Stokes equations. Comm Pure Appl Math, 2011, 64(9):1297-1304
[20] Lemarié-Rieusset P G. The Navier-Stokes Problem in the 21st Century. CRC Press, 2016
[21] Liu Q, Zhao J. Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces. J Math Anal Appl, 2014, 420(2):1301-1315
[22] Liu Q, Zhao J, Cui S. Existence and regularizing rate estimates of solutions to a generalized magnetohydrodynamic system in pseudomeasure spaces. Ann Mat Pura Appl, 2012, 191(2):293-309
[23] Miao C, Yuan B. On well-posedness of the Cauchy problem for MHD system in Besov spaces. Math Methods Appl Sci, 2009, 32(1):53-76
[24] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Commun Pure Appl Math, 1983, 36:635-664
[25] Sickel W. Smoothness spaces related to Morrey spaces -a survey. I. Eurasian Math J, 2012, 3(3):110-149
[26] Taylor M E. Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Commun Partial Differ Equ, 1992, 17:1407-1456
[27] Tran C, Yu X, Zhai Z. Note on solution regularity of the generalized nagnetohydrodynamic equations with partial dissipation. Nonlinear Anal, 2013, 85:43-51
[28] Wang Y. Asymptotic decay of solutions to 3D MHD equations. Nonlinear Anal, 2016, 132:115-125
[29] Wang Y. BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations. J Math Anal Appl, 2007, 328:1082-1086
[30] Wang Y, Wang K. Global well-posedness of the three dimensional magnetohydrodynamics equations. Nonlinear Anal RWA, 2014, 17:245-251
[31] Wu J. Generalized MHD equations. J Differ Equ, 2003, 195:284-312
[32] Wu J. Global regularity for a class of generalized magnetohydrodynamic equations. J Math Fluid Mech, 2011, 13:295-305
[33] Xiao W, Chen J, Fan D, Zhou X. Global well-posedness and long time decay of fractional Navier-Stokes equations in fourier Besov spaces. Abstr Appl Anal, 2014, Art:463639
[34] Xu X, Ye Z, Zhang Z. Remark on an improved regularity criterion for the 3D MHD equations. Appl Math Lett, 2015, 42:41-46
[35] Yuan J. Existence theorem and regularity criteria for the generalized MHD equations. Nonlinear Anal Real World Appl, 2010, 11(3):1640-1649
[36] Zhou Y. Reguality criteria for the generalized viscous MHD equations. Ann Inst H Poincaré, Anal Non Linéaire, 2007, 24:491-505
[37] Zhou Z, Gala S. Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z Angew Math Phys, 2010, 61:193-199
[38] Zhuan Y. Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations. Ann Mat Pura Appl, 2016, 195(4):1111-1121

WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES

WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES

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