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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 |
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 and Applied Analysis,
2019, 18
(2)
: 943-958.
doi: 10.3934/cpaa.2019046
![]()
References:
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H. Cabannes,
Theoretical Magnetofluid Dynamics, Applied Mathematics and Mechanics, New York, 1970. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [6] |
R. Courant and K. O. Friedrichs,
Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
T. T. Li,
Global Classical Solutions for Quasilinear Hyperbolic System,
John Wiley and Sons, 1994. |
| [19] |
T. T. Li and T. H. Qin,
Physics and Partial Differential Equations (in Chinese),
Higher Education Press, 2005. |
| [20] |
T. T. Li and W. C. Yu,
Boundary Value Problem for Quasilinear Hyperbolic Systems,
Duke University, 1985. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
T. Y. Zhang and X. Y. Zheng,
Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.
|
| [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. |
| [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. |
| [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. |
| [34] |
Y. X. Zheng, Systems of Conservation Laws: 2D Riemann Problems, 38 PNLDE, Bikhäuser, Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |
show all references
References:
| [1] |
H. Cabannes,
Theoretical Magnetofluid Dynamics, Applied Mathematics and Mechanics, New York, 1970. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [6] |
R. Courant and K. O. Friedrichs,
Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
T. T. Li,
Global Classical Solutions for Quasilinear Hyperbolic System,
John Wiley and Sons, 1994. |
| [19] |
T. T. Li and T. H. Qin,
Physics and Partial Differential Equations (in Chinese),
Higher Education Press, 2005. |
| [20] |
T. T. Li and W. C. Yu,
Boundary Value Problem for Quasilinear Hyperbolic Systems,
Duke University, 1985. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
T. Y. Zhang and X. Y. Zheng,
Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.
|
| [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. |
| [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. |
| [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. |
| [34] |
Y. X. Zheng, Systems of Conservation Laws: 2D Riemann Problems, 38 PNLDE, Bikhäuser, Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |
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