zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Two regularity criteria for the 3D MHD equations. (English) Zbl 1190.35046
Summary: This work establishes two regularity criteria for the 3D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction.

MSC:
35B65Smoothness and regularity of solutions of PDE
35B45A priori estimates for solutions of PDE
76W05Magnetohydrodynamics and electrohydrodynamics
References:
[1]Adams, R. A.: Sobolev spaces, (1975)
[2]Agapito, R.; Schonbek, M.: Non-uniform decay of MHD equations with and without magnetic diffusion, Comm. partial differential equations 32, 1791-1812 (2007) · Zbl 1132.35073 · doi:10.1080/03605300701318658
[3]Beale, J. T.; Kato, T.; Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. math. Phys. 94, 61-66 (1984) · Zbl 0573.76029 · doi:10.1007/BF01212349
[4]Da Veiga, H. Beirão: A new regularity class for the Navier – Stokes equations in rn, Chinese ann. Math. 16, 407-412 (1995) · Zbl 0837.35111
[5]Da Veiga, H. Beirão: On the smoothness of a class of weak solutions to the Navier – Stokes equations, J. math. Fluid mech. 2, 315-323 (2000) · Zbl 0972.35089 · doi:10.1007/PL00000955
[6]Berselli, L. C.; Galdi, G. P.: Regularity criteria involving the pressure for the weak solutions to the Navier – Stokes equations, Proc. amer. Math. soc. 130, 3585-3595 (2002) · Zbl 1075.35031 · doi:10.1090/S0002-9939-02-06697-2
[7]Caflisch, R.; Klapper, I.; Steele, G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. math. Phys. 184, 443-455 (1997) · Zbl 0874.76092 · doi:10.1007/s002200050067
[8]C. Cao, Sufficient conditions for the regularity to the 3D Navier – Stokes equations, Discrete Contin. Dyn. Syst. (Special issue), in press
[9]Cao, C.; Qin, J.; Titi, E.: Regularity criterion for solutions of three-dimensional turbulent channel flows, Comm. partial differential equations 33, 419-428 (2008) · Zbl 1151.76012 · doi:10.1080/03605300701454859
[10]Cao, C.; Titi, E.: Regularity criteria for the three-dimensional Navier – Stokes equations, Indiana univ. Math. J. 57, 2643-2662 (2008) · Zbl 1159.35053 · doi:10.1512/iumj.2008.57.3719
[11]Cao, C.; Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, (19 January 2009)
[12]Chae, D.: On the regularity conditions for the Navier – Stokes and related equations, Rev. mat. Iberoamericana 23, 371-384 (2007) · Zbl 1130.35100
[13]Chae, D.: Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity, J. funct. Anal. 254, 441-453 (2008) · Zbl 1163.35030 · doi:10.1016/j.jfa.2007.10.001
[14]Chae, D.; Lee, J.: Regularity criterion in terms of pressure for the Navier – Stokes equations, Nonlinear anal. 46, 727-735 (2001) · Zbl 1007.35064 · doi:10.1016/S0362-546X(00)00163-2
[15]Chae, D.; Choe, H. J.: Regularity of solutions to the Navier – Stokes equations, Electron. J. Differential equations 5, 1-7 (1999) · Zbl 0923.35117 · doi:emis:journals/EJDE/Volumes/1999/05/abstr.html
[16]Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability, (1961) · Zbl 0142.44103
[17]Chen, Q.; Miao, C.; Zhang, Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. math. Phys. 284, 919-930 (2008) · Zbl 1168.35035 · doi:10.1007/s00220-008-0545-y
[18]Córdoba, D.; Marliani, C.: Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D, Comm. pure appl. Math. 53, 512-524 (2000) · Zbl 1038.76060 · doi:10.1002/(SICI)1097-0312(200004)53:4<512::AID-CPA4>3.0.CO;2-R
[19]Duvaut, G.; Lions, J. -L.: Inéquations en thermoélasticité et magnétohydrodynamique, Arch. ration. Mech. anal. 46, 241-279 (1972) · Zbl 0264.73027 · doi:10.1007/BF00250512
[20]Escauriaza, L.; Seregin, G.; Šverák, V.: Backward uniqueness for parabolic equations, Arch. ration. Mech. anal. 169, 147-157 (2003) · Zbl 1039.35052 · doi:10.1007/s00205-003-0263-8
[21]Escauriaza, L.; Seregin, G.; Šverák, V.: L3,-solutions of the Navier – Stokes equations and backward uniqueness, Russian math. Surveys 58, 211-250 (2003) · Zbl 1064.35134 · doi:10.1070/RM2003v058n02ABEH000609
[22]Galdi, G. P.: An introduction to the mathematical theory of the Navier – Stokes equations, vols. I, II, (1994)
[23]Gibbon, J. D.; Ohkitani, K.: Evidence for singularity formation in a class of stretched solutions of the equations for ideal MHD, Fluid mech. Appl. 71, 295-304 (2002) · Zbl 1142.76510
[24]Hasegawa, A.: Self-organization processed in continuous media, Adv. phys. 34, 1-42 (1985)
[25]He, C.: New sufficient conditions for regularity of solutions to the Navier – Stokes equations, Adv. math. Sci. appl. 12, 535-548 (2002) · Zbl 1039.35077
[26]He, C.; Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. differential equations 238, 1-17 (2007) · Zbl 1220.35117 · doi:10.1016/j.jde.2007.03.023
[27]He, C.; Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations, J. differential equations 213, 235-254 (2005) · Zbl 1072.35154 · doi:10.1016/j.jde.2004.07.002
[28]He, C.; Xin, Z.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. funct. Anal. 227, 113-152 (2005) · Zbl 1083.35110 · doi:10.1016/j.jfa.2005.06.009
[29]Kozono, H.; Ogawa, T.; Taniuchi, Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242, 251-278 (2002) · Zbl 1055.35087 · doi:10.1007/s002090100332
[30]Kozono, H.; Yatsu, N.: Extension criterion via two-components of vorticity on strong solution to the 3D Navier – Stokes equations, Math. Z. 246, 55-68 (2003) · Zbl 1060.35105 · doi:10.1007/s00209-003-0576-1
[31]Kukavica, I.; Ziane, M.: One component regularity for the Navier – Stokes equations, Nonlinearity 19, 453-469 (2006) · Zbl 1149.35069 · doi:10.1088/0951-7715/19/2/012
[32]Kukavica, I.; Ziane, M.: Regularity of the Navier – Stokes equation in a thin periodic domain with large data, Discrete contin. Dyn. syst. 16, 67-86 (2006) · Zbl 1115.35098 · doi:10.3934/dcds.2006.16.67
[33]Ladyzhenskaya, O. A.: Mathematical theory of viscous incompressible flow, (1969) · Zbl 0184.52603
[34]Lei, Z.; Zhou, Y.: Bkm’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, (19 January 2009)
[35]Miao, C.; Yuan, B.: Well-posedness of the ideal MHD system in critical Besov spaces, Methods appl. Anal. 13, 89-106 (2006) · Zbl 1202.35173 · doi:euclid:maa/1175797482
[36]Miao, C.; Yuan, B.; Zhang, B.: Well-posedness for the incompressible magneto-hydrodynamic system, Math. methods appl. Sci. 30, 961-976 (2007) · Zbl 1115.76082 · doi:10.1002/mma.820
[37]Neustupa, J.; Penel, P.: Regularity of a suitable weak solution to the Navier – Stokes equations as a consequence of regularity of one velocity component, , 391-402 (1999) · Zbl 0953.35113
[38]Núñez, M.: Estimates on hyperdiffusive magnetohydrodynamics, Phys. D 183, 293-301 (2003) · Zbl 1044.76072 · doi:10.1016/S0167-2789(03)00173-8
[39]Ohkitani, K.: A note on regularity conditions on ideal magnetohydrodynamic equations, Phys. plasmas 13, 044504 (2006)
[40]Politano, H.; Pouquet, A.; Sulem, P. L.: Current and vorticity dynamics in three dimensional magnetohydrodynamic turbulence, Phys. plasmas 2, 2931-2939 (1995)
[41]Penel, P.; Pokorný, M.: Some new regularity criteria for the Navier – Stokes equations containing gradient of the velocity, Appl. math. 49, 483-493 (2004) · Zbl 1099.35101 · doi:10.1023/B:APOM.0000048124.64244.7e
[42]Pokorný, M.: On the result of he concerning the smoothness of solutions to the Navier – Stokes equations, Electron. J. Differential equations 10, 1-8 (2003) · Zbl 1014.35073 · doi:emis:journals/EJDE/Volumes/2003/11/abstr.html
[43]Schonbek, M. E.; Schonbek, T. P.; Süli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations, Math. ann. 304, 717-756 (1996) · Zbl 0846.35018 · doi:10.1007/BF01446316
[44]Seregin, G.; Šverák, V.: Navier – Stokes equations with lower bounds on the pressure, Arch. ration. Mech. anal. 163, 65-86 (2002) · Zbl 1002.35094 · doi:10.1007/s002050200199
[45]Sermange, M.; Temam, R.: Some mathematical questions related to the MHD equations, Comm. pure appl. Math. 36, 635-664 (1983) · Zbl 0524.76099 · doi:10.1002/cpa.3160360506
[46]Serrin, J.: On the interior regularity of weak solutions of the Navier – Stokes equations, Arch. ration. Mech. anal. 9, 187-195 (1962) · Zbl 0106.18302 · doi:10.1007/BF00253344
[47]Wu, J.: Viscous and inviscid magnetohydrodynamics equations, J. anal. Math. 73, 251-265 (1997)
[48]Wu, J.: Bounds and new approaches for the 3D MHD equations, J. nonlinear sci. 12, 395-413 (2002)
[49]Wu, J.: Generalized MHD equations, J. differential equations 195, 284-312 (2003)
[50]Wu, J.: Regularity results for weak solutions of the 3D MHD equations, Discrete contin. Dyn. syst. 10, 543-556 (2004) · Zbl 1055.76062 · doi:10.3934/dcds.2004.10.543
[51]Wu, J.: Regularity criteria for the generalized MHD equations, Comm. partial differential equations 33, 285-306 (2008) · Zbl 1134.76068 · doi:10.1080/03605300701382530
[52]Yuan, B.: Regularity criterion of weak solutions to the MHD system based on vorticity and electric current in negative index Besov spaces, Adv. math. (China) 37, 451-458 (2008)
[53]Zhang, Z.; Chen, Q.: Regularity criterion via two components of vorticity on weak solutions to the Navier – Stokes equations in R3, J. differential equations 216, 470-481 (2005) · Zbl 1091.35064 · doi:10.1016/j.jde.2005.06.001
[54]Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier – Stokes equations in a generic domain, Math. ann. 328, 173-192 (2004) · Zbl 1054.35062 · doi:10.1007/s00208-003-0478-x
[55]Zhou, Y.: Regularity criteria for the generalized viscous MHD equations, Ann. inst. H. Poincaré anal. Non linéaire 24, 491-505 (2007) · Zbl 1130.35110 · doi:10.1016/j.anihpc.2006.03.014 · doi:numdam:AIHPC_2007__24_3_491_0