XU Xin-Ying . A BLOW-UP CRITERION FOR 3-D NON-RESISTIVE COMPRESSIBLE HEAT-CONDUCTIVE MAGNETOHYDRODYNAMIC EQUATIONS#br# WITH INITIAL VACUUM[J]. Acta mathematica scientia, Series B, 2012 , 32(5) : 1883 -1900 . DOI: 10.1016/S0252-9602(12)60147-7

[1] Beale J T, Kato T, Majda A. Remarks on the breakdown of smooth soutions for the 3-D Euler equations. Commun Math Phys, 1984, 94: 61–66

[2] Bergh J, L¨ofst¨om J. Interpolation Spaces. New York: Spinger-Verlag, 1976

[3] Cabannes H. Theoretical Magnetofluiddynamics. New York: Academic Press, 1970

[4] Caflisch R, Klapper I, Steele G. Remarks on singularities, dimensional and energy dissipation for ideal hydrodynamics and MHD. Commun Math Phys, 1997, 184(2): 443–455

[5] Cao C, Wu J. Two regularity criteria for the 3D MHD equations. J Differ Equ, 2010, 248(9): 2263–2274

[6] Chen G -Q, Wang D. Global solution of nonlinear magnetohydrodynamics with large initial data. J Diff Eqns, 2002, 182: 344–376

[7] Chen G -Q,Wang D. Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z Angew Math Phys, 2003, 54: 608–632

[8] Chen Q, Miao C, Zhang Z. The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations. Commun Math Phys, 2007, 275: 861–875

[9] Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluids. J Math Pures Appl, 2004, 83: 2243–275

[10] Ducomet B, Feireisl E. The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Commun Math Phys, 2006, 266: 595–629

[11] Fan J, Jiang S. Blow-up criteria for the Navier-Stokes equations of compressible fluids. J Hyper Differ Eqns, 2008, 5(1): 167–185

[12] Fan J, Jiang S, Ou Y. A blow-up criteria for compressible viscous heat-conducting flows. Ann Inst H Poincar´e Anal Non Lin´eaire, 2010, 27(1): 337–350

[13] Fan J, Jiang S, Nakamura G. Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun Math Phys, 2007, 270: 691–708

[14] Fan J, Yu W. Global variational solutions to the compressible magnetohydrodynamic equations. Nonlinear Anal, 2008, 69: 3637–3660

[15] J. Fan, W. Yu, Strong solution to the compressible MHD equations with vacuum. Nonlinear Anal Real World Appl, 2009, 10: 392–409

[16] Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004

[17] Feireisl E, Novotny A, Petzeltov´a H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3: 358–392

[18] Freidberg J P. Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev Modern Physics, 1982, 54(3): 801–902

[19] He C, Xin Z. On the regularity of weak solutions to the magnetohydrodynamic equations. J Differ Equ, 2005, 213: 235–254

[20] Hu X, Wang D. Global solutions to the three-dimensional full compressilbe magnetohydrodynamic flows. Commun Math Phys, 2008, 283: 255–284

[21] Hu X, Wang D. Compactness of weak solutions to the three-dimensional compressilbe magnetohydrodynamic equations. J Diff Eqns, 2008, 245: 2176–2198

[22] Hu X, Wang D. Low mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J Math Anal, 2009, 41: 127–1294

[23] Hu X, Wang D. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch Rational Mech Anal, 2010, 197(1): 203–238

[24] Huang X D, Xin Z -P. A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci China Math, 2010, 53(3): 671–686

[25] Huang X D, Li J, Xin Z -P. Blowup criterion for the compressible flows with vacuum states. Commun Math Phys, 2010, 301: 23–35

[26] Huang X D, Li J, Xin Z -P. Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J Math Anal, 2011, 43(4): 1872–1886

[27] Jiang S, Ju Q -C, Li F -C. Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun Math Phys, 2010, 297: 371–400

[28] Jiang S, Ou Y. A blow-up criterion for compressible viscous heat-conductive flows. Acta Math Sci, 2010, 30B(6): 1851–1864

[29] Jiu Q S, Niu D J. Mathematical results related to a two-dimensional magneto-hydrodynamic equations. Acta Math Sci, 2006, 26B(4): 744–756

[30] Kawashima S. Smooth global solutions for two-dimensional equations of electromagneto-fluid dynamics. Japan J Appl Math, 1984, 1: 207–222

[31] Kawashima S, Okada M. Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc Japan Acad Ser A Math Sci, 1982, 58: 384–387

[32] Kawashima S, Shizuta Y. Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. Tsukuba J Math, 1986, 10: 131–149

[33] Kawashima S, Shizuta Y. Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II. Proc Japan Acad Ser A, 1986, 62: 181–184

[34] Kazhikhov V, Shelukhin V V. Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas. J Appl Math Mech, 1977, 41: 273–282

[35] Landau L D, Lifshitz E M, Pitaevskii L P. Electrodynamics of Continuous Media. 2nd ed. London: Butterworth-Heinemann, 1999

[36] Lei Z, Masmoudi N, Zhou Y. Remarks on the blowup criteria for Oldroyd models. J Differ Equ, 2010, 248: 328–341

[37] Lions P -L. Mathematical Topics in Fluid Dynamics, Vol 2. Compressible Models. Oxford: Oxford Science Publication, 1996

[38] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heatconductive gases. J Math Kyoto Univ, 1980, 20: 67–104

[39] Matsumura A, Nishida T. Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun Math Phys, 1983, 89: 445–464

[40] Novotn´y A, Stra?skraba I. Introduction to the Mathematical Theory of Compressible Flow. Oxford: Oxford University Press, 2004

[41] Ponce G. Remarks on a paper “Remarks on the breakdown of smooth solutions for the 3-D Euler equations”. Commun Math Phys, 1985, 98: 349–353

[42] Rozanova O. Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations. J Differ Equ, 2008, 245(7): 1762–1774

[43] Rozanova O. Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy//Hyperbolic Problems: Theory, Numerics and Applications. Proc Sympos Appl Math, 67, Part 2. Providence, RI: Amer Math Soc, 2009: 911–917

[44] Umeda T, Kawashima S, Shizuta Y. On the decay of solutions to the linearized equations of electromagnetofuid dynamic. Japan J Appl Math, 1984, 1: 435–457

[45] Volpert A I, Khudiaev S I. on the cauchy problem for composite systems of nonlinear equations. Mat Sbornik, 1972, 87: 504–528

[46] Sun Y, Wang C, Zhang Z. A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J Math Pures Anal, 201, 95: 36–47

[47] Wang D. Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J Appl Math, 2003, 63: 1424–1441

[48] Woods L C. Priciples of Magnetoplasma Dynamics. Oxford: Oxford University Press, 1987

[49] Wu J. Bounds and new appraoches for the 3D MHD equations. J Nonlinear Sci, 2002, 12(4): 395–413

[50] Wu J. Regularity results for weak solutions of the 3D MHD equations. Disc Cont Dyna Sys, 2004, 10: 543–556

[51] Xin Z -P. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51: 229–240

[52] Xu X Y, Zhang J W. A blow-up criterion for 3-D non-resistive compressible magnetohydrodynamic equations with initial vacuum. Nonlinear Anal Real World Appl, 2011, 12(6): 3442–3451

[53] Xu X Y, Zhang J W. A blow-up criterion for 3-D compressible magnetohydrodynamic equations with vacuum. Math Mod Meth Appl Sci, 2012, 22(2): 1150010

[54] Zhang J W, Jiang S, Xie F. Global weak solutions of an initial boundary value problem for screw pinches in plasma physics. Math Mod Meth Appl Sci, 2009, 19: 833–875

[55] Zhang J W, Xie F. Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J Differ Equ, 2008, 245: 1853–1882

[56] Zhang J W, Zhao J N. Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics. Comm Math Sci, 2010, 8(4): 835–850

[57] Zhou Y. Remarks on regularities for the 3D MHD equations. Disc Cout Dyna Sys, 2005, 12: 881–886