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March  2019, 18(2): 943-958. doi: 10.3934/cpaa.2019046

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

Jianjun Chen 1, and  Geng Lai 2,,

1. 

Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, China
2. 

Department of Mathematics, Shanghai University, Shanghai, 200444, China

* Corresponding author

Received  April 2018 Revised  May 2018 Published  October 2018

Fund Project: Supported by NSF of China (11301326) and the grant of the first-class Discipline of Universities in Shanghai.

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.

Keywords: MHD equations, self-similar flow, semi-hyperbolic patch, Goursat problem, characteristic decomposition.
Mathematics Subject Classification: Primary: 35L65, 35J70, 35R35; Secondary: 35J65.
Citation: Jianjun Chen, Geng Lai. Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 943-958. doi: 10.3934/cpaa.2019046
References:
[1]

H. Cabannes, Theoretical Magnetofluid Dynamics, Applied Mathematics and Mechanics, New York, 1970.  Google Scholar

[2]

S. Canic and B. L. Keyfitz, Quasi-one-dimensional Riemann problems and their role in selfsimilar two-dimensional problems, Arch. Ration. Mech. Anal., 144 (1998), 233-258.  doi: 10.1007/s002050050117.  Google Scholar

[3]

G. Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Ann. of Math. (2), 171 (2010), 1067-1182.  doi: 10.4007/annals.2010.171.1067.  Google Scholar

[4]

J. J. Chen, G. Lai, and W. C. Sheng, Characteristic decompositions and interaction of rarefaction waves to the two-dimensional compressible Euler equations in magnetohydrodynamics, submitted. Google Scholar

[5]

X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59 (2010), 231-256.   Google Scholar

[6]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[7]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.  Google Scholar

[8]

K. G. Guderley, Considerations on the structure of mixed subsonicsupersonic flow patterns, Air Material Command Tech. Report F-TR-2168-ND, ATI 22780, GS-AAF-Wright Field 39, U.S. Wright-Patterson Air Force Base, Dayton, OH, 1947. Google Scholar

[9]

Y. Hu, J. Q. Li and W. C. Sheng, Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations, Z. Angew. Math. Phys., 63 (2012), 1021-1046.  doi: 10.1007/s00033-012-0203-2.  Google Scholar

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Y. B Hu and G. D. Wang, Semi-hyperbolic pathces of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.  doi: 10.1016/j.jde.2014.05.020.  Google Scholar

[11]

E. H. Kim and C. Tsikkou, Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions, Discrete Contin. Dyn. Syst., 37 (2017), 6257-6289.  doi: 10.3934/dcds.2017271.  Google Scholar

[12]

G. Lai and W. C. Sheng, Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics, J. Math. Pures Appl., 104 (2015), 179-206.  doi: 10.1016/j.matpur.2015.02.005.  Google Scholar

[13]

G. Lai and W. C. Sheng, Two-dimensional centered wave flow patches to the Guderley Mach reflection configurations for steady flow in gas dynamics, J. Hyperbolic Differ. Equ., 13 (2016), 107-128.  doi: 10.1142/S0219891616500028.  Google Scholar

[14]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.  Google Scholar

[15]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interaction of rarefaction waves of 2-D Euler equations, J. Differential Equations., 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[16]

J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys., 267 (2006), 1-12.  doi: 10.1007/s00220-006-0033-1.  Google Scholar

[17]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.  Google Scholar

[18]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic System, John Wiley and Sons, 1994.  Google Scholar

[19]

T. T. Li and T. H. Qin, Physics and Partial Differential Equations (in Chinese), Higher Education Press, 2005.  Google Scholar

[20]

T. T. Li and W. C. Yu, Boundary Value Problem for Quasilinear Hyperbolic Systems, Duke University, 1985.  Google Scholar

[21]

D. Serre, Multi-dimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191 (2008), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[22]

B. W. Skews and J. T. Ashworth, The physical nature of weak shock wave reflection, J. Fluid Mech., 542 (2005), 105-114.  doi: 10.1017/S0022112005006543.  Google Scholar

[23]

K. Song and Y. X. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380.  doi: 10.3934/dcds.2009.24.1365.  Google Scholar

[24]

K. Song, Q. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.  Google Scholar

[25]

A. M. Tesdall and J. K. Hunter, Self-similar solutions for weak shock reflection, SIAM J. Appl. Math., 63 (2002), 42-61.  doi: 10.1137/S0036139901383826.  Google Scholar

[26]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.  doi: 10.1137/060660758.  Google Scholar

[27]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math.(2015), 68 (2008), 1360-1377.  doi: 10.1137/070698567.  Google Scholar

[28]

Q. Wang and K. Song, The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas, Discrete Contin. Dyn. Syst., 36 (2016), 1661-1675.  doi: 10.3934/dcds.2016.36.1661.  Google Scholar

[29]

A. Zakharian, M. Brio, J. K. Hunter and G. Webb, The von Neumann paradox in weak shock reflection, J. Fluid Mech., 422 (2000), 193-205.  doi: 10.1017/S0022112000001609.  Google Scholar

[30]

T. Y. Zhang and X. Y. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.   Google Scholar

[31]

T. Y. Zhang and X. Y. Zheng, The structure of solutions near a sonic line in gas dynamics via the pressure gradient equation, J. Math. Anal. Appl., 443 (2016), 39-56.  doi: 10.1016/j.jmaa.2016.04.002.  Google Scholar

[32]

T. Zhang and X. Y. Zheng, Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Appl. Math., 21 (1990), 593-630.  doi: 10.1137/0521032.  Google Scholar

[33]

Y. X. Zheng, Absorption of characteristics by sonic curve of the two-dimensional Euler equations, Discrete Contin. Dyn. Syst., 23 (2009), 605-616.  doi: 10.3934/dcds.2009.23.605.  Google Scholar

[34]

Y. X. Zheng, Systems of Conservation Laws: 2D Riemann Problems, 38 PNLDE, Bikhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

show all references

References:
[1]

H. Cabannes, Theoretical Magnetofluid Dynamics, Applied Mathematics and Mechanics, New York, 1970.  Google Scholar

[2]

S. Canic and B. L. Keyfitz, Quasi-one-dimensional Riemann problems and their role in selfsimilar two-dimensional problems, Arch. Ration. Mech. Anal., 144 (1998), 233-258.  doi: 10.1007/s002050050117.  Google Scholar

[3]

G. Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Ann. of Math. (2), 171 (2010), 1067-1182.  doi: 10.4007/annals.2010.171.1067.  Google Scholar

[4]

J. J. Chen, G. Lai, and W. C. Sheng, Characteristic decompositions and interaction of rarefaction waves to the two-dimensional compressible Euler equations in magnetohydrodynamics, submitted. Google Scholar

[5]

X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59 (2010), 231-256.   Google Scholar

[6]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[7]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.  Google Scholar

[8]

K. G. Guderley, Considerations on the structure of mixed subsonicsupersonic flow patterns, Air Material Command Tech. Report F-TR-2168-ND, ATI 22780, GS-AAF-Wright Field 39, U.S. Wright-Patterson Air Force Base, Dayton, OH, 1947. Google Scholar

[9]

Y. Hu, J. Q. Li and W. C. Sheng, Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations, Z. Angew. Math. Phys., 63 (2012), 1021-1046.  doi: 10.1007/s00033-012-0203-2.  Google Scholar

[10]

Y. B Hu and G. D. Wang, Semi-hyperbolic pathces of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.  doi: 10.1016/j.jde.2014.05.020.  Google Scholar

[11]

E. H. Kim and C. Tsikkou, Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions, Discrete Contin. Dyn. Syst., 37 (2017), 6257-6289.  doi: 10.3934/dcds.2017271.  Google Scholar

[12]

G. Lai and W. C. Sheng, Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics, J. Math. Pures Appl., 104 (2015), 179-206.  doi: 10.1016/j.matpur.2015.02.005.  Google Scholar

[13]

G. Lai and W. C. Sheng, Two-dimensional centered wave flow patches to the Guderley Mach reflection configurations for steady flow in gas dynamics, J. Hyperbolic Differ. Equ., 13 (2016), 107-128.  doi: 10.1142/S0219891616500028.  Google Scholar

[14]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.  Google Scholar

[15]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interaction of rarefaction waves of 2-D Euler equations, J. Differential Equations., 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[16]

J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys., 267 (2006), 1-12.  doi: 10.1007/s00220-006-0033-1.  Google Scholar

[17]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.  Google Scholar

[18]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic System, John Wiley and Sons, 1994.  Google Scholar

[19]

T. T. Li and T. H. Qin, Physics and Partial Differential Equations (in Chinese), Higher Education Press, 2005.  Google Scholar

[20]

T. T. Li and W. C. Yu, Boundary Value Problem for Quasilinear Hyperbolic Systems, Duke University, 1985.  Google Scholar

[21]

D. Serre, Multi-dimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191 (2008), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[22]

B. W. Skews and J. T. Ashworth, The physical nature of weak shock wave reflection, J. Fluid Mech., 542 (2005), 105-114.  doi: 10.1017/S0022112005006543.  Google Scholar

[23]

K. Song and Y. X. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380.  doi: 10.3934/dcds.2009.24.1365.  Google Scholar

[24]

K. Song, Q. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.  Google Scholar

[25]

A. M. Tesdall and J. K. Hunter, Self-similar solutions for weak shock reflection, SIAM J. Appl. Math., 63 (2002), 42-61.  doi: 10.1137/S0036139901383826.  Google Scholar

[26]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.  doi: 10.1137/060660758.  Google Scholar

[27]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math.(2015), 68 (2008), 1360-1377.  doi: 10.1137/070698567.  Google Scholar

[28]

Q. Wang and K. Song, The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas, Discrete Contin. Dyn. Syst., 36 (2016), 1661-1675.  doi: 10.3934/dcds.2016.36.1661.  Google Scholar

[29]

A. Zakharian, M. Brio, J. K. Hunter and G. Webb, The von Neumann paradox in weak shock reflection, J. Fluid Mech., 422 (2000), 193-205.  doi: 10.1017/S0022112000001609.  Google Scholar

[30]

T. Y. Zhang and X. Y. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.   Google Scholar

[31]

T. Y. Zhang and X. Y. Zheng, The structure of solutions near a sonic line in gas dynamics via the pressure gradient equation, J. Math. Anal. Appl., 443 (2016), 39-56.  doi: 10.1016/j.jmaa.2016.04.002.  Google Scholar

[32]

T. Zhang and X. Y. Zheng, Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Appl. Math., 21 (1990), 593-630.  doi: 10.1137/0521032.  Google Scholar

[33]

Y. X. Zheng, Absorption of characteristics by sonic curve of the two-dimensional Euler equations, Discrete Contin. Dyn. Syst., 23 (2009), 605-616.  doi: 10.3934/dcds.2009.23.605.  Google Scholar

[34]

Y. X. Zheng, Systems of Conservation Laws: 2D Riemann Problems, 38 PNLDE, Bikhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

Figure 1.  Pseudosonic curve. (Ⅰ) Keldysh type; (Ⅱ) Tricomi type
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Figure 2.  A semi-hyperbolic patch
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Figure 3.  Left: small Goursat problem; right: global solution
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