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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

Baoquan Yuan 1, and  Xiaokui Zhao 2,

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.
Keywords: Full compressible MHD system, smooth solution, blowup..
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B44, 76W0.
Citation: Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

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