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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

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.

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. HuJ. 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. LiZ. 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. LiT. 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. SongQ. 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. TesdallR. 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. TesdallR. 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. ZakharianM. BrioJ. 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. HuJ. 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. LiZ. 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. LiT. 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. SongQ. 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. TesdallR. 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. TesdallR. 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. ZakharianM. BrioJ. 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
Figure 2.  A semi-hyperbolic patch
Figure 3.  Left: small Goursat problem; right: global solution
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