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

Local regularity of the magnetohydrodynamics equations near the curved boundary

Jae-Myoung  Kim 1,

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.
Keywords: Magnetohydrodynamics equations, local regularity condition., suitable weak solutions.
Mathematics Subject Classification: Primary: 35Q30, 35K1.
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, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.  Google Scholar

[2]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar

[3]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, (French) [Inéquations en thermoélasticité et magnétohydrodynamique], Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[4]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.  Google Scholar

[5]

K. Kang and J. Lee, Interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations, J. Differential Equations, 247 (2009), 2310-2330. doi: 10.1016/j.jde.2009.07.016.  Google Scholar

[6]

K. Kang and J.-M. Kim, Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations, J. Funct. Anal., 266 (2014), 99-120. doi: 10.1016/j.jfa.2013.09.007.  Google Scholar

[7]

J. Kim and M. Kim, Local regularity of the Navier-Stokes equations near the curved boundary, J. Math. Anal. Appl., 363 (2010), 161-173. doi: 10.1016/j.jmaa.2009.08.015.  Google Scholar

[8]

O. A. Ladyžhenskaya and V. A. Solonnikov, Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid (in Russian), Proceedings of V.A. Steklov Mathematical Institute, 59 (1960), 115-173 .  Google Scholar

[9]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[10]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[11]

V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552.  Google Scholar

[12]

V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42.  Google Scholar

[13]

G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 204-223, 317; translation in J. Math. Sci. (N. Y.) 115 (2003), 2820-2831. doi: 10.1023/A:1023330105200.  Google Scholar

[14]

V. Vyalov, Partial regularity of solutions to the magnetohydrodynamic equations, J. Math. Sci. (N. Y.), 150 (2008), 1771-1786. 5002-5009. doi: 10.1007/s10958-008-0095-z.  Google Scholar

[15]

V. Vyalov, On the boundary regularity of weak solutions to the MHD system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 41, 18-53, 234; translation in J. Math. Sci. (N. Y.) 178 (2011), 243-264. doi: 10.1007/s10958-011-0545-x.  Google Scholar

[16]

V. Vyalov, On the local smoothness of weak solutions to the MHD system near the boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 397 (2011), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 42, 5-19, 172. doi: 10.1007/s10958-012-0950-9.  Google Scholar

[17]

V. Vyalov and T. Shilkin, Estimates of solutions to the perturbed Stokes system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 410 (2013), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 43, 5-24, 187.  Google Scholar

[18]

W. Wang and Z. Zhang, On the interior regularity criteria for suitable weak solutions of the Magneto-hydrodynamics equations, SIAM J. Math. Anal., 45 (2013), 2666-2677. doi: 10.1137/120879646.  Google Scholar

show all references

References:
[1]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.  Google Scholar

[2]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar

[3]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, (French) [Inéquations en thermoélasticité et magnétohydrodynamique], Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[4]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.  Google Scholar

[5]

K. Kang and J. Lee, Interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations, J. Differential Equations, 247 (2009), 2310-2330. doi: 10.1016/j.jde.2009.07.016.  Google Scholar

[6]

K. Kang and J.-M. Kim, Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations, J. Funct. Anal., 266 (2014), 99-120. doi: 10.1016/j.jfa.2013.09.007.  Google Scholar

[7]

J. Kim and M. Kim, Local regularity of the Navier-Stokes equations near the curved boundary, J. Math. Anal. Appl., 363 (2010), 161-173. doi: 10.1016/j.jmaa.2009.08.015.  Google Scholar

[8]

O. A. Ladyžhenskaya and V. A. Solonnikov, Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid (in Russian), Proceedings of V.A. Steklov Mathematical Institute, 59 (1960), 115-173 .  Google Scholar

[9]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[10]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[11]

V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552.  Google Scholar

[12]

V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42.  Google Scholar

[13]

G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 271 (2000), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 204-223, 317; translation in J. Math. Sci. (N. Y.) 115 (2003), 2820-2831. doi: 10.1023/A:1023330105200.  Google Scholar

[14]

V. Vyalov, Partial regularity of solutions to the magnetohydrodynamic equations, J. Math. Sci. (N. Y.), 150 (2008), 1771-1786. 5002-5009. doi: 10.1007/s10958-008-0095-z.  Google Scholar

[15]

V. Vyalov, On the boundary regularity of weak solutions to the MHD system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 41, 18-53, 234; translation in J. Math. Sci. (N. Y.) 178 (2011), 243-264. doi: 10.1007/s10958-011-0545-x.  Google Scholar

[16]

V. Vyalov, On the local smoothness of weak solutions to the MHD system near the boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 397 (2011), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 42, 5-19, 172. doi: 10.1007/s10958-012-0950-9.  Google Scholar

[17]

V. Vyalov and T. Shilkin, Estimates of solutions to the perturbed Stokes system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 410 (2013), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 43, 5-24, 187.  Google Scholar

[18]

W. Wang and Z. Zhang, On the interior regularity criteria for suitable weak solutions of the Magneto-hydrodynamics equations, SIAM J. Math. Anal., 45 (2013), 2666-2677. doi: 10.1137/120879646.  Google Scholar

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