October  2020, 19(10): 4973-4994. doi: 10.3934/cpaa.2020223

Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  February 2020 Revised  June 2020 Published  July 2020

Fund Project: The author is partially supported by NSFC grant 11801460

This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding Du's et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in $ \mathbb{R}^2 $ when the initial density and the initial magnetic field are decay not too show at infinity, and Ji's et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.

Citation: Yongfu Wang. Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4973-4994. doi: 10.3934/cpaa.2020223
References:
[1]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

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

[3]

L. L. Du and Y. F. Wang, Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence, Commun. Math. Sci., 12 (2014), 1427-1435.  doi: 10.4310/CMS.2014.v12.n8.a3.

[4]

L. L. Du and Y. F. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.

[5]

J. S. Fan and W. H. 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.

[6]

X. P. Hu and D. H. 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.

[7]

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

[8]

X. D. HuangJ. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal, 43 (2011), 1872-1886.  doi: 10.1137/100814639.

[9]

X. D. HuangJ. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commun. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.

[10]

R. H. Ji and Y. F. Wang, Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations, Discrete Contin. Dyn. Syst., 39 (2019), 1117-1133.  doi: 10.3934/dcds.2019047.

[11]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[12]

S. Jiang and P. Zhang, Axisymmetric solutions of the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pure Appl., 82 (2003), 949-973.  doi: 10.1016/S0021-7824(03)00015-1.

[13]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$, Proc. Symp. Pure Math., 45 (1986), 1-7.  doi: 10.1090/pspum/045.2.

[14]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagneto-fluid dynamics, Jpn. J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.

[15]

H. L. Li, X. Y. Xu and J. W. 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]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 4 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[17]

J. Li and Z. P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Anal. Partial Differ. Equ., 5 (2019), Art. 7. doi: 10.1007/s40818-019-0064-5.

[18] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998. 
[19]

B. Q. Lü and B. Huang, On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.  doi: 10.1088/0951-7715/28/2/509.

[20]

B. Q. LüX. D. Shi and X. Y. 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.

[21]

B. Q. LüZ. H. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 107 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[22]

A. Novotny and I. Straŝkraba, Introduction to The Mathematical Theory of Compressible Flow, Oxford University Press, 2004.

[23]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equation, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[24]

Y. Z. SunC. Wang and Z. F. 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.

[25]

Y. Z. Sun and Z. F. Zhang, A Blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations, Sci. China Math., 54 (2011), 105-116.  doi: 10.1007/s11425-010-4045-0.

[26]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[27]

Y. F. Wang and S. Li, Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 18 (2014), 23-33.  doi: 10.1016/j.nonrwa.2014.01.006.

[28]

Y. H. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572.  doi: 10.1016/j.aim.2013.07.018.

[29]

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

[30]

X. Y. Xu and J. W. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Meth. Appl. Sci., 22 (2012), Art. 1150010, 23 pp. doi: 10.1142/S0218202511500102.

show all references

References:
[1]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[2]

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

[3]

L. L. Du and Y. F. Wang, Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence, Commun. Math. Sci., 12 (2014), 1427-1435.  doi: 10.4310/CMS.2014.v12.n8.a3.

[4]

L. L. Du and Y. F. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.

[5]

J. S. Fan and W. H. 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.

[6]

X. P. Hu and D. H. 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.

[7]

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

[8]

X. D. HuangJ. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal, 43 (2011), 1872-1886.  doi: 10.1137/100814639.

[9]

X. D. HuangJ. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commun. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.

[10]

R. H. Ji and Y. F. Wang, Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations, Discrete Contin. Dyn. Syst., 39 (2019), 1117-1133.  doi: 10.3934/dcds.2019047.

[11]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[12]

S. Jiang and P. Zhang, Axisymmetric solutions of the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pure Appl., 82 (2003), 949-973.  doi: 10.1016/S0021-7824(03)00015-1.

[13]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$, Proc. Symp. Pure Math., 45 (1986), 1-7.  doi: 10.1090/pspum/045.2.

[14]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagneto-fluid dynamics, Jpn. J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.

[15]

H. L. Li, X. Y. Xu and J. W. 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]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 4 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[17]

J. Li and Z. P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Anal. Partial Differ. Equ., 5 (2019), Art. 7. doi: 10.1007/s40818-019-0064-5.

[18] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998. 
[19]

B. Q. Lü and B. Huang, On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.  doi: 10.1088/0951-7715/28/2/509.

[20]

B. Q. LüX. D. Shi and X. Y. 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.

[21]

B. Q. LüZ. H. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 107 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.

[22]

A. Novotny and I. Straŝkraba, Introduction to The Mathematical Theory of Compressible Flow, Oxford University Press, 2004.

[23]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equation, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[24]

Y. Z. SunC. Wang and Z. F. 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.

[25]

Y. Z. Sun and Z. F. Zhang, A Blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations, Sci. China Math., 54 (2011), 105-116.  doi: 10.1007/s11425-010-4045-0.

[26]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[27]

Y. F. Wang and S. Li, Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations, Nonlinear Anal. Real World Appl., 18 (2014), 23-33.  doi: 10.1016/j.nonrwa.2014.01.006.

[28]

Y. H. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572.  doi: 10.1016/j.aim.2013.07.018.

[29]

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

[30]

X. Y. Xu and J. W. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Meth. Appl. Sci., 22 (2012), Art. 1150010, 23 pp. doi: 10.1142/S0218202511500102.

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