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Regularity criteria for the generalized viscous MHD equations. (English) Zbl 1130.35110
Generalized 3D viscous MHD equations are considered. Navier-Stokes and standard MHD equations are obtained for particular values of parameters. A regularity criteria, similar to Serrin’s regularity class for Navier-Stokes equations is given, and a smooth solution on $(0,T]$ interval is obtained. A weak solution is considered, which becomes strong if the corresponding vorticity field belongs to a particular Hölder space. The main tools are interpolation and inequalities in fractional Sobolev spaces.

35Q35PDEs in connection with fluid mechanics
35Q30Stokes and Navier-Stokes equations
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI Numdam EuDML
[1] Da Veiga, H. Beirão: A new regularity class for the Navier -- Stokes equations in rn. Chinese ann. Math. 16, 407-412 (1995)
[2] Da Veiga, H. Beirão: Vorticity and smoothness in viscous flows. Int. math. Ser. (N.Y.) 2, 61-67 (2002) · Zbl 1183.76666
[3] Da Veiga, H. Beirão; Berselli, L. C.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Differential integral equations 15, No. 3, 345-356 (2002) · Zbl 1014.35072
[4] Caffarelli, L.; Kohn, R.; Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier -- Stokes equations. Comm. pure appl. Math. 35, 771-831 (1982) · Zbl 0509.35067
[5] Caflisch, R.; Klapper, I.; Steele, G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. math. Phys. 184, No. 2, 443-455 (1997) · Zbl 0874.76092
[6] Chemin, J. Y.: Perfect incompressible fluids. Oxford lecture series in mathematics and its applications 14 (1998)
[7] Constantin, P.; Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier -- Stokes equations. Indiana univ. Math. J. 42, 775-789 (1993) · Zbl 0837.35113
[8] Duoandikoetxea, J.: Fourier analysis. Graduate studies in mathematics 29 (2001) · Zbl 0969.42001
[9] He, C.: On partial regularity for weak solutions to the Navier -- Stokes equations. J. funct. Anal. 211, No. 1, 153-162 (2004) · Zbl 1062.35065
[10] He, C.; Xin, Z.: On the regularity of solutions to the magnetohydrodynamic equations. J. differential equations 213, No. 2, 235-254 (2005) · Zbl 1072.35154
[11] Sermange, M.; Temam, R.: Some mathematical questions related to the MHD equations. Comm. pure appl. Math. 36, No. 5, 635-664 (1983) · Zbl 0524.76099
[12] Serrin, J.: On the interior regularity of weak solutions of the Navier -- Stokes equations. Arch. rational mech. Anal. 9, 187-195 (1962) · Zbl 0106.18302
[13] Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton mathematical series 30 (1970) · Zbl 0207.13501
[14] Tian, G.; Xin, Z.: Gradient estimation on Navier -- Stokes equations. Comm. anal. Geom. 7, 221-257 (1999) · Zbl 0939.35139
[15] Wu, J.: Generalized MHD equations. J. differential equations 195, 284-312 (2003) · Zbl 1057.35040
[16] Wu, J.: Bounds and new approaches for the 3D MHD equations. J. nonlinear sci. 12, No. 4, 395-413 (2002) · Zbl 1029.76062
[17] Zhou, Y.: A new regularity criterion for the Navier -- Stokes equations in terms of the gradient of one velocity component. Method appl. Anal. 9, No. 4, 563-578 (2002) · Zbl 1166.35359
[18] Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier -- Stokes equations in a generic domain. Math. ann. 328, No. 1 -- 2, 173-192 (2004) · Zbl 1054.35062
[19] Zhou, Y.: A new regularity criterion for the Navier -- Stokes equations in terms of the direction of vorticity. Monatsh. math. 144, 251-257 (2005) · Zbl 1072.35148
[20] Zhou, Y.: A new regularity criterion for weak solutions to the Navier -- Stokes equations. J. math. Pures appl. (9) 84, No. 11, 1496-1514 (2005) · Zbl 1092.35081
[21] Zhou, Y.: Remarks on regularities for the 3D MHD equations. Discrete contin. Dynam. syst. 12, No. 5, 881-886 (2005) · Zbl 1068.35117
[22] Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier -- Stokes equations in RN. Z. angew. Math. phys. 57, 384-392 (2006) · Zbl 1099.35091