# American Institute of Mathematical Sciences

March  2016, 15(2): 507-517. doi: 10.3934/cpaa.2016.15.507

## Local regularity of the magnetohydrodynamics equations near the curved boundary

 1 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea

Received  April 2015 Revised  October 2015 Published  January 2016

We study a local regularity condition for a suitable weak solutions of the magnetohydrodynamics equations near the curved boundary.
Citation: Jae-Myoung  Kim. Local regularity of the magnetohydrodynamics equations near the curved boundary. Communications on Pure & Applied Analysis, 2016, 15 (2) : 507-517. doi: 10.3934/cpaa.2016.15.507
##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations,, \emph{Comm. Pure Appl. Math.}, 35 (1982), 771. doi: 10.1002/cpa.3160350604. [2] P. A. Davidson, An Introduction to Magnetohydrodynamics,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511626333. [3] G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, (French) [In\'equations en thermo\'elasticit\'e et magn\'etohydrodynamique], 46 (1972), 241. [4] C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 227 (2005), 113. doi: 10.1016/j.jfa.2005.06.009. [5] K. Kang and J. Lee, Interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations,, \emph{J. Differential Equations}, 247 (2009), 2310. doi: 10.1016/j.jde.2009.07.016. [6] K. Kang and J.-M. Kim, Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 266 (2014), 99. doi: 10.1016/j.jfa.2013.09.007. [7] J. Kim and M. Kim, Local regularity of the Navier-Stokes equations near the curved boundary,, \emph{J. Math. Anal. Appl.}, 363 (2010), 161. doi: 10.1016/j.jmaa.2009.08.015. [8] O. A. Ladyžhenskaya and V. A. Solonnikov, Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid (in Russian),, \emph{Proceedings of V.A. Steklov Mathematical Institute}, 59 (1960), 115. [9] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 241. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. [10] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635. doi: 10.1002/cpa.3160360506. [11] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations,, \emph{Pacific J. Math.}, 66 (1976), 535. [12] V. Scheffer, The Navier-Stokes equations on a bounded domain,, \emph{Comm. Math. Phys.}, 73 (1980), 1. [13] G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 271 (2000), 204. doi: 10.1023/A:1023330105200. [14] V. Vyalov, Partial regularity of solutions to the magnetohydrodynamic equations,, \emph{J. Math. Sci. (N. Y.)}, 150 (2008), 1771. doi: 10.1007/s10958-008-0095-z. [15] V. Vyalov, On the boundary regularity of weak solutions to the MHD system,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)} \textbf{385} (2010), 385 (2010), 18. doi: 10.1007/s10958-011-0545-x. [16] V. Vyalov, On the local smoothness of weak solutions to the MHD system near the boundary,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 397 (2011), 5. doi: 10.1007/s10958-012-0950-9. [17] V. Vyalov and T. Shilkin, Estimates of solutions to the perturbed Stokes system,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 410 (2013), 5. [18] W. Wang and Z. Zhang, On the interior regularity criteria for suitable weak solutions of the Magneto-hydrodynamics equations,, \emph{SIAM J. Math. Anal.}, 45 (2013), 2666. doi: 10.1137/120879646.

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##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations,, \emph{Comm. Pure Appl. Math.}, 35 (1982), 771. doi: 10.1002/cpa.3160350604. [2] P. A. Davidson, An Introduction to Magnetohydrodynamics,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511626333. [3] G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, (French) [In\'equations en thermo\'elasticit\'e et magn\'etohydrodynamique], 46 (1972), 241. [4] C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 227 (2005), 113. doi: 10.1016/j.jfa.2005.06.009. [5] K. Kang and J. Lee, Interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations,, \emph{J. Differential Equations}, 247 (2009), 2310. doi: 10.1016/j.jde.2009.07.016. [6] K. Kang and J.-M. Kim, Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations,, \emph{J. Funct. Anal.}, 266 (2014), 99. doi: 10.1016/j.jfa.2013.09.007. [7] J. Kim and M. Kim, Local regularity of the Navier-Stokes equations near the curved boundary,, \emph{J. Math. Anal. Appl.}, 363 (2010), 161. doi: 10.1016/j.jmaa.2009.08.015. [8] O. A. Ladyžhenskaya and V. A. Solonnikov, Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid (in Russian),, \emph{Proceedings of V.A. Steklov Mathematical Institute}, 59 (1960), 115. [9] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 241. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. [10] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 635. doi: 10.1002/cpa.3160360506. [11] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations,, \emph{Pacific J. Math.}, 66 (1976), 535. [12] V. Scheffer, The Navier-Stokes equations on a bounded domain,, \emph{Comm. Math. Phys.}, 73 (1980), 1. [13] G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 271 (2000), 204. doi: 10.1023/A:1023330105200. [14] V. Vyalov, Partial regularity of solutions to the magnetohydrodynamic equations,, \emph{J. Math. Sci. (N. Y.)}, 150 (2008), 1771. doi: 10.1007/s10958-008-0095-z. [15] V. Vyalov, On the boundary regularity of weak solutions to the MHD system,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)} \textbf{385} (2010), 385 (2010), 18. doi: 10.1007/s10958-011-0545-x. [16] V. Vyalov, On the local smoothness of weak solutions to the MHD system near the boundary,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 397 (2011), 5. doi: 10.1007/s10958-012-0950-9. [17] V. Vyalov and T. Shilkin, Estimates of solutions to the perturbed Stokes system,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 410 (2013), 5. [18] W. Wang and Z. Zhang, On the interior regularity criteria for suitable weak solutions of the Magneto-hydrodynamics equations,, \emph{SIAM J. Math. Anal.}, 45 (2013), 2666. doi: 10.1137/120879646.
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