March  2014, 7(1): 195-203. doi: 10.3934/krm.2014.7.195

Blowup of smooth solutions to the full compressible MHD system with compact density

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan

2. 

School of Mathematics and Information Science, Henan polytechnic University, Jiaozuo 454000, Henan, China

Received  July 2013 Revised  September 2013 Published  December 2013

This paper studies the blowup of smooth solutions to the full compressible MHD system with zero resistivity on $\mathbb{R}^{d}$, $d\geq 1$. We obtain that the smooth solutions to the MHD system will blow up in finite time, if the initial density is compactly supported.
Citation: Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic and Related Models, 2014, 7 (1) : 195-203. doi: 10.3934/krm.2014.7.195
References:
[1]

D.-F. Bian and B.-L. Guo, Blow-up of smooth solutions to the isentropic compressible MHD equations, to appear in Applicable Analysis, 2013. doi: 10.1080/00036811.2013.766324.

[2]

Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826. doi: 10.1016/j.jmaa.2005.08.005.

[3]

D. Du, J. Li and K. Zhang, Blow-up of smooth solutions to the Navier-Stokes equations for compressible isothermal fluids, Commun. Math. Sci., 11 (2013), 541-546. doi: 10.4310/CMS.2013.v11.n2.a11.

[4]

O. Rozanova, Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy, in Hyperbolic Problems: Theory, Numerics and Applications (eds. E. Tadmor, J.-G. Liu and A. E. Tzavaras), Proc. Sympos. Appl. Math., 67, Part 2, Amer. Math. Soc., Providence, RI, 2009, 911-917. doi: 10.1090/psapm/067.2/2605286.

[5]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.

[6]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.

[7]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Commun. Math. Sci., 321 (2013), 529-541. doi: 10.1007/s00220-012-1610-0.

show all references

References:
[1]

D.-F. Bian and B.-L. Guo, Blow-up of smooth solutions to the isentropic compressible MHD equations, to appear in Applicable Analysis, 2013. doi: 10.1080/00036811.2013.766324.

[2]

Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826. doi: 10.1016/j.jmaa.2005.08.005.

[3]

D. Du, J. Li and K. Zhang, Blow-up of smooth solutions to the Navier-Stokes equations for compressible isothermal fluids, Commun. Math. Sci., 11 (2013), 541-546. doi: 10.4310/CMS.2013.v11.n2.a11.

[4]

O. Rozanova, Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy, in Hyperbolic Problems: Theory, Numerics and Applications (eds. E. Tadmor, J.-G. Liu and A. E. Tzavaras), Proc. Sympos. Appl. Math., 67, Part 2, Amer. Math. Soc., Providence, RI, 2009, 911-917. doi: 10.1090/psapm/067.2/2605286.

[5]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.

[6]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.

[7]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Commun. Math. Sci., 321 (2013), 529-541. doi: 10.1007/s00220-012-1610-0.

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