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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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