doi: 10.3934/dcdsb.2018318

A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity

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

Received  October 2017 Published  January 2019

Fund Project: Supported by Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2018jcyjAX0049), the Postdoctoral Science Foundation of Chongqing (No. xm2017015), and China Postdoctoral Science Foundation (Nos. 2018T110936, 2017M610579)

We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies $ \|\nabla{\bf{u}}\|_{L^{2}(0,T;L^\infty)}<\infty $. Our method relies upon the delicate energy estimates and elliptic estimates.

Citation: Xin Zhong. A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018318
References:
[1]

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

[2]

X. Cai and Y. Sun, Blowup criteria for strong solutions to the compressible Navier-Stokes equations with variable viscosity, Nonlinear Anal. Real World Appl., 29 (2016), 1-18. doi: 10.1016/j.nonrwa.2015.10.007.

[3]

Y. ChenX. Hou and L. Zhu, A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Meth. Appl. Sci., 40 (2017), 5526-5538. doi: 10.1002/mma.4407.

[4]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[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]

E. FeireislA. Novotný and Y. Sun, A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 301 (2014), 219-239. doi: 10.1007/s00205-013-0697-6.

[8]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.

[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. HuangJ. Li and Y. Wang, Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316. doi: 10.1007/s00205-012-0577-5.

[13]

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

[14]

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

[15]

X. D. HuangJ. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.

[16]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics (Ph. D. thesis), Kyoto University, 1983.

[17] O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
[18]

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.

[19] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅱ: compressible models, Oxford University Press, Oxford, 1998.
[20]

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.

[21] A. Novotný and I. Stra$\check{s}$kraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.
[22]

A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805; Corrigendum, Discrete Contin. Dyn. Syst., 35 (2015), 1387-1390. doi: 10.3934/dcds.2013.33.3791.

[23]

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.

[24]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1.

[25]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, American Mathematical Society, Providence, R. I., 2006. doi: 10.1090/cbms/106.

[26]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213; Correction, Ann. Mat. Pura Appl., 132 (1983), 399-400. doi: 10.1007/BF01760990.

[27]

A. I. Vol'pert and S. I. Khudiaev, On the Cauchy problem for composite systems nonlinear equations, Mat. Sb, 87 (1972), 504-528.

[28]

H. Wen and C. 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. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.

[30]

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.

[31]

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

[32]

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

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, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.

[2]

X. Cai and Y. Sun, Blowup criteria for strong solutions to the compressible Navier-Stokes equations with variable viscosity, Nonlinear Anal. Real World Appl., 29 (2016), 1-18. doi: 10.1016/j.nonrwa.2015.10.007.

[3]

Y. ChenX. Hou and L. Zhu, A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Meth. Appl. Sci., 40 (2017), 5526-5538. doi: 10.1002/mma.4407.

[4]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[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]

E. FeireislA. Novotný and Y. Sun, A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 301 (2014), 219-239. doi: 10.1007/s00205-013-0697-6.

[8]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.

[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. HuangJ. Li and Y. Wang, Serrin-type blowup criterion for full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316. doi: 10.1007/s00205-012-0577-5.

[13]

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

[14]

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

[15]

X. D. HuangJ. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.

[16]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics (Ph. D. thesis), Kyoto University, 1983.

[17] O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
[18]

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.

[19] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅱ: compressible models, Oxford University Press, Oxford, 1998.
[20]

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.

[21] A. Novotný and I. Stra$\check{s}$kraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.
[22]

A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805; Corrigendum, Discrete Contin. Dyn. Syst., 35 (2015), 1387-1390. doi: 10.3934/dcds.2013.33.3791.

[23]

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.

[24]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1.

[25]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, American Mathematical Society, Providence, R. I., 2006. doi: 10.1090/cbms/106.

[26]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213; Correction, Ann. Mat. Pura Appl., 132 (1983), 399-400. doi: 10.1007/BF01760990.

[27]

A. I. Vol'pert and S. I. Khudiaev, On the Cauchy problem for composite systems nonlinear equations, Mat. Sb, 87 (1972), 504-528.

[28]

H. Wen and C. 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. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.

[30]

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.

[31]

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

[32]

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

[1]

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

[2]

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

[3]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[4]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399

[5]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327

[6]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[7]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[8]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

[9]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828

[10]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[11]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[12]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

[13]

Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585

[14]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771

[15]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

[16]

John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174

[17]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677

[18]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465

[19]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[20]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

2017 Impact Factor: 0.972

Article outline

[Back to Top]