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 & 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 \emph{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,, \emph{J. Math. Anal. Appl.}, 320 (2006), 819. 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,, \emph{Commun. Math. Sci.}, 11 (2013), 541. 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 \emph{Hyperbolic Problems: Theory, (2009), 911. 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,, \emph{J. Differential Equations}, 245 (2008), 1762. 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,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 229. Google Scholar

[7]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations,, \emph{Commun. Math. Sci.}, 321 (2013), 529. 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 \emph{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,, \emph{J. Math. Anal. Appl.}, 320 (2006), 819. 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,, \emph{Commun. Math. Sci.}, 11 (2013), 541. 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 \emph{Hyperbolic Problems: Theory, (2009), 911. 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,, \emph{J. Differential Equations}, 245 (2008), 1762. 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,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 229. Google Scholar

[7]

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

[1]

Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084

[2]

Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure & Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161

[3]

Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283

[4]

Paola Trebeschi. On the slightly compressible MHD system in the half-plane. Communications on Pure & Applied Analysis, 2004, 3 (1) : 97-113. doi: 10.3934/cpaa.2004.3.97

[5]

Baoquan Yuan. Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2211-2219. doi: 10.3934/dcds.2013.33.2211

[6]

Ming He, Jianwen Zhang. Global cylindrical solution to the compressible MHD equations in an exterior domain. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1841-1865. doi: 10.3934/cpaa.2009.8.1841

[7]

Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068

[8]

Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785

[9]

Boling Guo, Haiyang Huang. Smooth solution of the generalized system of ferro-magnetic chain. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 729-740. doi: 10.3934/dcds.1999.5.729

[10]

Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237

[11]

Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583

[12]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002

[13]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005

[14]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

[15]

Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure & Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959

[16]

Rinaldo M. Colombo, Graziano Guerra. A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case. Networks & Heterogeneous Media, 2016, 11 (2) : 313-330. doi: 10.3934/nhm.2016.11.313

[17]

Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835

[18]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167

[19]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711

[20]

Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]