In this paper, we establish new regularity criteria for the three-dimensional (3D) viscous incompressible magnetohydrodynamic (MHD) equations. It is proved that if the solution of the MHD equations satisfies or , then the regularity of the solution on (0, T), where u3, j3 and ω3 are the third component of velocity u, current density ∇ × b and vorticity ∇ × u, respectively. These results give new improvements of regularity theory of weak solutions.
Topics
Navier Stokes equations
REFERENCES
1.
Beirão da Veiga
, H.
, “A new regularity class for the Navier-Stokes equations in
,” Chin. Ann. Math. Ser. B
16
, 407
–412
(1995
), see https://camath.fudan.edu.cn/cambcn/ch/reader/create_pdf.aspx?file_no=16B401&flag=1.2.
Cao
, C.
, Qin
, J.
, and Titi
, E. S.
, “Regularity criterion for solutions of three-dimensional turbulent channel flows
,” Commun. Partial Differ. Equations
33
(3
), 419
–428
(2008
).3.
Cao
, C.
and Titi
, E. S.
, “Regularity criteria for the three-dimensional Navier–Stokes equations
,” Indiana Univ. Math. J.
57
, 2643
–2661
(2008
).4.
Cao
, C.
and Titi
, E. S.
, “Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor
,” Arch. Ration. Mech. Anal.
202
(3
), 919
–932
(2011
).5.
Duvaut
, G.
and Lions
, J. L.
, “Inéquations en thermoélasticité et magnétohydrodynamique
,” Arch. Ration. Mech. Anal.
46
, 241
–279
(1972
).6.
Escauriaza
, L.
, Seregin
, G.
, and Šverák
, V.
, “Backward uniqueness for parabolic equations
,” Arch. Ration. Mech. Anal.
169
, 147
–157
(2003
).7.
Guo
, Z.
, Li
, Y.
, and Skalák
, Z.
, “On conditional regularity for the MHD equations via partial components
,” J. Math. Fluid Mech.
23
(3
), 51
(2021
).8.
He
, C.
and Xin
, Z.
, “Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations
,” J. Funct. Anal.
227
, 113
–152
(2005
).9.
Hopf
, E.
, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet
,” Math. Nachr.
4
, 213
–231
(1950
).10.
Jia
, X.
and Zhou
, Y.
, “On regularity criteria for the 3D incompressible MHD equations involving one velocity component
,” J. Math. Fluid Mech.
18
, 187
–206
(2016
).11.
Kukavica
, I.
and Ziane
, M.
, “One component regularity for the Navier–Stokes equations
,” Nonlinearity
19
, 453
–469
(2006
).12.
Leray
, J.
, “Sur le mouvement d’un liquide visqueux emplissant l’espace
,” Acta Math.
63
, 193
–248
(1934
).13.
Neustupa
, J.
and Penel
, P.
, “Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component
,” in Applied Nonlinear Analysis
, edited by A.
Sequeira
, H. B.
da Veiga
, and J. H.
Videman
(Springer, Boston, MA
, 2002
), pp. 391
–402
.14.
Prodi
, G.
, “Un teorema di unicità per le equazioni di Navier-Stokes
,” Ann. Mat. Pura Appl.
48
, 173
–182
(1959
).15.
Robinson
, J. C.
, Rodrigo
, J. L.
, and Sadowski
, W.
, The Three-Dimensional Navier-Stokes Equations: Classical Theory
, Cambridge Studies in Advanced Mathematics Vol. 157
(Cambridge University Press
, Cambridge
, 2016
).16.
Sermange
, M.
and Temam
, R.
, “Some mathematical questions related to the MHD equations
,” Commun. Pure Appl. Math.
36
, 635
–664
(1983
).17.
Serrin
, J.
, “On the interior regularity of weak solutions of the Navier–Stokes equations
,” Arch. Ration. Mech. Anal.
9
, 187
–195
(1962
).18.
Yamazaki
, K.
, “Regularity criteria of MHD system involving one velocity and one current density component
,” J. Math. Fluid Mech.
16
, 551
–570
(2014
).19.
Yamazaki
, K.
, “Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components
,” J. Math. Phys.
55
, 031505
(2014
).20.
Yamazaki
, K.
, “Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component
,” Nonlinear Anal.: Theory, Methods Appl.
135
, 73
–83
(2016
).21.
Yamazaki
, K.
, “On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces
,” Bull. Sci. Math.
140
, 575
–614
(2016
).22.
Zhang
, Z.
, “Remarks on the global regularity criteria for the 3D MHD equations via two components
,” Z. Angew. Math. Phys.
66
, 977
–987
(2015
).23.
Zhang
, Z.
, “Refined regularity criteria for the MHD system involving only two components of the solution
,” Appl. Anal.
96
, 2130
–2139
(2017
).24.
Zhang
, Z.
and Zhang
, Y.
, “An improved regularity criteria for the MHD system based on two components of the solution
,” Appl. Math.
66
, 451
–460
(2021
).25.
Zhou
, Y.
, “Remarks on regularities for the 3D MHD equations
,” Discrete Contin. Dyn. Syst., A
12
, 881
–886
(2005
).26.
Zhou
, Y.
and Pokorný
, M.
, “On the regularity of the solutions of the Navier–Stokes equations via one velocity component
,” Nonlinearity
23
, 1097
–1107
(2010
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
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