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Local regularity of the magnetohydrodynamics equations near the curved boundary
1. | Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea |
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. |
[2] |
P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511626333. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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 . |
[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. |
[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. |
[11] |
V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. |
[12] |
V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511626333. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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 . |
[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. |
[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. |
[11] |
V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. |
[12] |
V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys., 73 (1980), 1-42. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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