doi: 10.3934/dcdsb.2019209

Singularity formation to the two-dimensional non-baratropic 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  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-baratropic 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-baratropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete & Continuous Dynamical Systems - B, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, Indiana Univ. Math. J.(preprint), 2019.Google Scholar

[25]

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

[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. Google Scholar

[27]

X. Zhong, Strong solutions to the Cauchy problem of the two-dimensional non-baratropic non-resistive magnetohydrodynamic equations with zero heat conduction, https://arXiv.org/abs/1801.07589 doi: 10.1063/1.4906902. Google Scholar

[28]

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

[29]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

X. Zhong, On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction, Indiana Univ. Math. J.(preprint), 2019.Google Scholar

[25]

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

[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. Google Scholar

[27]

X. Zhong, Strong solutions to the Cauchy problem of the two-dimensional non-baratropic non-resistive magnetohydrodynamic equations with zero heat conduction, https://arXiv.org/abs/1801.07589 doi: 10.1063/1.4906902. Google Scholar

[28]

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

[29]

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

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