LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

doi:10.1016/S0252-9602(17)30056-5

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (4): 1033-1047.doi: 10.1016/S0252-9602(17)30056-5

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

Jae-Myoung KIM

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

收稿日期:2016-03-04 修回日期:2016-05-23 出版日期:2017-08-25 发布日期:2017-08-25
作者简介:Jae-Myoung KIM,E-mail:cauchy02@naver.com
基金资助:
Jae-Myoung Kim is also partly supported by BK21 PLUS SNU Mathematical Sciences Division and Basic Science Research Program through the National Research Foundation of Korea (NRF) (NRF-2016R1D1A1B03930422).

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

Jae-Myoung KIM

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

Received:2016-03-04 Revised:2016-05-23 Online:2017-08-25 Published:2017-08-25
About author:Jae-Myoung KIM,E-mail:cauchy02@naver.com
Supported by:
Jae-Myoung Kim is also partly supported by BK21 PLUS SNU Mathematical Sciences Division and Basic Science Research Program through the National Research Foundation of Korea (NRF) (NRF-2016R1D1A1B03930422).

摘要/Abstract

摘要：

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are Hölder continuous near boundary provided that the scaled mixed L_x,t^p,q -norm of the velocity vector field with 3/p + 2/q ≤ 2, 2 < q < ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_x,t^p,q with 1 ≤ 3/p + 2/q ≤ 3/2, 3 < p < ∞, the Hausdorff dimension of its singular set is no greater than max{p,q}(3/p + 2/q -1).

关键词: local regularity criteria, suitable weak solution, MHD equations

Abstract:

Key words: local regularity criteria, suitable weak solution, MHD equations

Jae-Myoung KIM. LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS[J]. 数学物理学报(英文版), 2017, 37(4): 1033-1047.

Jae-Myoung KIM. LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS[J]. Acta mathematica scientia,Series B, 2017, 37(4): 1033-1047.

参考文献

[1] Bae H-O, Kang K, Kim M-H. Local regularity criteria of the Navier-Stokes equations with slip boundary conditions. J Korean Math Soc, 2016, 53(3):597-621
[2] Caffarelli L, Kohn R, Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm Pure Appl Math, 1982, 35(6):771-831
[3] Davidson P A. An Introduction to Magnetohydrodynamics. Cambridge:Cambridge University Press, 2001
[4] Duvaut G, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique (French). Arch Rational Mech Anal, 1972, 46(6):241-279
[5] Fan J, Jiang S, Nakamura G, Zhou Y. Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J Math Fluid Mech, 2011, 13(4):557-571
[6] Fan J, Jia X, Nakamura G, Zhou Y. On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z Angew Math Phys, 2015, 66(4):1695-1706
[7] Gu X. Regularity criteria for suitable weak solutions to the four dimensional incompressible magnetohydrodynamic equations near boundary. J Differ Equ, 2015, 259(4):1354-1378
[8] Giga Y, Sohr H. Abstract L^p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J Funct Anal, 1991, 102(1):72-94
[9] Gustafson S, Kang K, Tsai T-P. Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary. J Differ Equ, 2006, 226(2):594-618
[10] He C, Xin Z. On the regularity of weak solutions to the magnetohydrodynamic equations. J Differ Equ, 2005, 213(2):235-254
[11] Jia X, Zhou Y. Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of 3×3 mixture matrices. Nonlinearity, 2015, 28(9):3289-3307
[12] Jia X, Zhou Y, On regularity criteria for the 3D incompressible MHD equations involving one velocity component. J Math Fluid Mech, 2016, 18(1):187-206
[13] Kang K, Kim J M. Regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space. J Differ Equ, 2016, 253(2):764-794
[14] Kang K, Kim J-M. Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations. J Funct Anal, 2014, 266(1):99-120
[15] Ladyžhenskaya O A, Solonnikov V A. Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid. Proceedings of VA Steklov Mathematical Institute, 1960, 59(1):115-173(in Russian)
[16] Robinson J C, Sadowski W. On the dimension of the singular set of solutions to the Navier-Stokes equations. Comm Math Phys, 2012, 309(2):497-506
[17] Sadowski W. A remark on the box-counting dimension of the singular set for the Navier-Stokes equations. Commun Math Sci, 2013, 11(2):597-602
[18] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36(5):635-664
[19] Serrin J. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch Rational Mech Anal, 1962, 9:187-195
[20] Shimada R. On the L_p-L_q maximal regularity for Stokes equations with Robin boundary condition in a bounded domain. Math Methods Appl Sci, 2007, 30(3):257-289
[21] Vyalov V, Shilkin T. On the boundary regularity of weak solutions to the MHD system. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI), 2010, 385:18-53; translation in J Math Sci (NY), 2011, 178(3):243-264
[22] Vyalov V. On the local smoothness of weak solutions to the MHD system near the boundary. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI), 2011, 3975-19; translation in J Math Sci (NY), 2012, 185(5):659-667
[23] Vyalov V, Shilkin T. Estimates of solutions to the perturbed Stokes system. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI), 2013, 410:5-24; translation in J Math Sci (NY), 2013, 195(1):1-12
[24] Wang W, Zhang Z. On the interior regularity criteria for suitable weak solutions of the Magnetohydrodynamics equations. SIAM J Math Anal, 2013, 45(5):2666-2677
[25] Zhou Y. Remarks on regularities for the 3D MHD equations. Discrete Contin Dyn Syst, 2005, 12(5):881-886
[26] Zhou Y. Regularity criteria for the 3D MHD equations in terms of the pressure. Int J Non Linear Mech, 2006, 41(10):1174-1180
[27] Zhou Y, Fan J. Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math, 2012, 24(4):691-708

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

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