March  2020, 25(3): 1083-1096. doi: 10.3934/dcdsb.2019209

Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  October 2018 Published  March 2020 Early access  September 2019

Fund Project: Supported by Fundamental Research Funds for the Central Universities (No. XDJK2019B031), Natural Science Foundation of Chongqing (No. cstc2018jcyjAX0049), the Postdoctoral Science Foundation of Chongqing (No. xm2017015), and China Postdoctoral Science Foundation (Nos. 2018T110936, 2017M610579).

We are concerned with the singularity formation of strong solutions to the two-dimensional (2D) non-barotropic non-resistive compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. It is showed that the strong solution exists globally if the density and the magnetic field as well as the pressure are bounded from above. Our method relies on critical Sobolev inequalities of logarithmic type.

Citation: Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209
References:
[1]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[2]

J. FanF. Li and G. Nakamura, A blow-up criterion to the 2D full compressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 38 (2015), 2073-2080.  doi: 10.1002/mma.3205.

[3]

J. FanF. Li and G. Nakamura, A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1757-1766.  doi: 10.3934/dcdsb.2018079.

[4]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.

[5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. 
[6]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[7]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008.

[8]

G. HongX. HouH. Peng and C. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 49 (2017), 2409-2441.  doi: 10.1137/16M1100447.

[9]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.

[10]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[11]

X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Comm. Math. Phys., 324 (2013), 147-171.  doi: 10.1007/s00220-013-1791-1.

[12]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[13]

X. D. Huang and Y. Wang, L continuation principle to the non-barotropic non-resistive magnetohydrodynamic equations without heat conductivity, ath. Methods Appl. Sci., 39 (2016), 4234-4245.  doi: 10.1002/mma.3860.

[14]

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamic, PhD thesis, Kyoto University, 1983.

[15]

H. LiX. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[16]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

L. LuY. Chen and B. Huang, Blow-up criterion for two-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows, Nonlinear Anal., 139 (2016), 55-74.  doi: 10.1016/j.na.2016.02.021.

[18]

B. LüX. Shi and X. Xu, Global well-posedness and large time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum, Indiana Univ. Math. J., 65 (2016), 925-975.  doi: 10.1512/iumj.2016.65.5813.

[19]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.

[20]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226.  doi: 10.1016/j.nonrwa.2013.09.020.

[21]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[22]

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

[23]

X. Zhong, A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3249-3264.  doi: 10.3934/dcdsb.2018318.

[24]

X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, Indiana Univ. Math. J., 68 (2019), 1379–1407.

[25]

X. Zhong, Singularity formation of the non-barotropic compressible magnetohydrodynamic equations without heat conductivity, Taiwanese J. Math., (2019), 26 pages. doi: 10.11650/tjm/190701.

[26]

X. Zhong, On local strong solutions to the 2D Cauchy problem of the compressible non-resistive magnetohydrodynamic equations with vacuum, J. Dynam. Differential Equations, (2019) 1–22. doi: 10.1007/s10884-019-09740-7.

[27]

X. Zhong, Strong solutions to the Cauchy problem of the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction, J. Differential Equations, (2019). doi: 10.1063/1.4906902.

[28]

X. Zhong, Singularity formation to the Cauchy problem of the two-dimensional non-barotropic magnetohydrodynamic equations without heat conductivity, https://arXiv.org/abs/1801.10036

[29]

X. Zhong, Singularity formation to the two-dimensional non-barotropic magnetohydrodynamic equations without heat conductivity in a bounded domain, submitted for publication.

show all references

References:
[1]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[2]

J. FanF. Li and G. Nakamura, A blow-up criterion to the 2D full compressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 38 (2015), 2073-2080.  doi: 10.1002/mma.3205.

[3]

J. FanF. Li and G. Nakamura, A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1757-1766.  doi: 10.3934/dcdsb.2018079.

[4]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.

[5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. 
[6]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[7]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008.

[8]

G. HongX. HouH. Peng and C. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 49 (2017), 2409-2441.  doi: 10.1137/16M1100447.

[9]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.

[10]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[11]

X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Comm. Math. Phys., 324 (2013), 147-171.  doi: 10.1007/s00220-013-1791-1.

[12]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.

[13]

X. D. Huang and Y. Wang, L continuation principle to the non-barotropic non-resistive magnetohydrodynamic equations without heat conductivity, ath. Methods Appl. Sci., 39 (2016), 4234-4245.  doi: 10.1002/mma.3860.

[14]

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamic, PhD thesis, Kyoto University, 1983.

[15]

H. LiX. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[16]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

L. LuY. Chen and B. Huang, Blow-up criterion for two-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows, Nonlinear Anal., 139 (2016), 55-74.  doi: 10.1016/j.na.2016.02.021.

[18]

B. LüX. Shi and X. Xu, Global well-posedness and large time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum, Indiana Univ. Math. J., 65 (2016), 925-975.  doi: 10.1512/iumj.2016.65.5813.

[19]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.

[20]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226.  doi: 10.1016/j.nonrwa.2013.09.020.

[21]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[22]

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

[23]

X. Zhong, A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3249-3264.  doi: 10.3934/dcdsb.2018318.

[24]

X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, Indiana Univ. Math. J., 68 (2019), 1379–1407.

[25]

X. Zhong, Singularity formation of the non-barotropic compressible magnetohydrodynamic equations without heat conductivity, Taiwanese J. Math., (2019), 26 pages. doi: 10.11650/tjm/190701.

[26]

X. Zhong, On local strong solutions to the 2D Cauchy problem of the compressible non-resistive magnetohydrodynamic equations with vacuum, J. Dynam. Differential Equations, (2019) 1–22. doi: 10.1007/s10884-019-09740-7.

[27]

X. Zhong, Strong solutions to the Cauchy problem of the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction, J. Differential Equations, (2019). doi: 10.1063/1.4906902.

[28]

X. Zhong, Singularity formation to the Cauchy problem of the two-dimensional non-barotropic magnetohydrodynamic equations without heat conductivity, https://arXiv.org/abs/1801.10036

[29]

X. Zhong, Singularity formation to the two-dimensional non-barotropic magnetohydrodynamic equations without heat conductivity in a bounded domain, submitted for publication.

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