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January  2018, 17(1): 127-142. doi: 10.3934/cpaa.2018008

The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas

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

* Corresponding author

Received  January 2017 Revised  June 2017 Published  September 2017

Fund Project: Supported by NSF of China (11371240) and Shanghai Municipal Education Commission of Scientific Research Innovation Project (11ZZ84)

This paper is concerned with the Euler equations in the magnetogasdynamics for generalized Chaplygin gas. The global solutions to the Riemann problems of the Euler equations in the magnetogasdynamics for generalized Chaplygin gas are obtained constructively by using phase plane analysis method. The formation of delta shock wave is studied as magnetic field vanishes. The limit behaviors of the Riemann solutions as magnetic field vanishes are also obtained.

Citation: Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008
References:
[1]

H. J. Cheng and H. C. Yang, Delta shock waves in chromatography equations, J. Math. Anal. Appl., 380 (2011), 475-485.  doi: 10.1016/j.jmaa.2011.04.002.  Google Scholar

[2]

G. Q. Chen and H. L. Liu, Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, SIAM J.Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar

[3]

S. Chaplygin, On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1-121.   Google Scholar

[4]

L. H. GuoW. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431-458.  doi: 10.3934/cpaa.2010.9.431.  Google Scholar

[5]

Y. B. Hu and W. C. Sheng, The Riemann problem of conservation laws in magnetogasdynamics, Commun.Pure Appl.Anal., 12 (2013), 755-769.  doi: 10.3934/cpaa.2013.12.755.  Google Scholar

[6]

T. von Karman, Compressibility effects in aerodynamics, J. Aeronaut. Sci., 8 (1941), 337-365.  doi: 10.2514/2.7046.  Google Scholar

[7]

G. LaiW. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523.  doi: 10.3934/dcds.2011.31.489.  Google Scholar

[8]

J. Q. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.  doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar

[9] T. T. Li and T. H. Qin, Physics and Partial Differential Equations (in Chinese), Higher Education Press, 2005.   Google Scholar
[10]

Y. J. Liu and W. H. Sun, Riemann problem and wave interactions in Magnetogasdynamics, J. Math. Anal. Appl., 397 (2013), 454-466.  doi: 10.1016/j.jmaa.2012.07.064.  Google Scholar

[11]

R. Singh and L. P. Singh., Solution of the Riemann Problem in magnetogasdynamics, International Journal of Non-Linear Mechanics, 67 (2014), 326-330.  doi: 10.1016/j.ijnonlinmec.2014.10.004.  Google Scholar

[12]

T. R. Sekhar and V. D. Sharma, Riemann Problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl., 11 (2010), 619-636.  doi: 10.1016/j.nonrwa.2008.10.036.  Google Scholar

[13]

D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[14]

C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Appl. Math. Lett., 24 (2011), 1124-1129.  doi: 10.1016/j.aml.2011.01.038.  Google Scholar

[15]

Z. Shao, Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish, Mathematics, 2015, arXiv: 1503.08382. Google Scholar

[16]

V. M. Shelkovich, The Riemann problem admitting δ, δ'-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500.  doi: 10.1016/j.jde.2006.08.003.  Google Scholar

[17]

W. C. ShengG. J. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Anal. Real World Appl., 22 (2015), 115-128.  doi: 10.1016/j.nonrwa.2014.08.007.  Google Scholar

[18]

W. C. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (1999), 654.  doi: 10.1090/memo/0654.  Google Scholar

[19]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci., 6 (1939), 399-407.  doi: 10.1016/B978-0-12-398277-3.50005-1.  Google Scholar

[20]

D. TanT. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.  Google Scholar

[21]

G. D. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar

[22]

G. Yin and W. C. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. Appl., 12 (2009), 1-12.  doi: 10.1016/j.jmaa.2009.01.075.  Google Scholar

show all references

References:
[1]

H. J. Cheng and H. C. Yang, Delta shock waves in chromatography equations, J. Math. Anal. Appl., 380 (2011), 475-485.  doi: 10.1016/j.jmaa.2011.04.002.  Google Scholar

[2]

G. Q. Chen and H. L. Liu, Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, SIAM J.Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar

[3]

S. Chaplygin, On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1-121.   Google Scholar

[4]

L. H. GuoW. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431-458.  doi: 10.3934/cpaa.2010.9.431.  Google Scholar

[5]

Y. B. Hu and W. C. Sheng, The Riemann problem of conservation laws in magnetogasdynamics, Commun.Pure Appl.Anal., 12 (2013), 755-769.  doi: 10.3934/cpaa.2013.12.755.  Google Scholar

[6]

T. von Karman, Compressibility effects in aerodynamics, J. Aeronaut. Sci., 8 (1941), 337-365.  doi: 10.2514/2.7046.  Google Scholar

[7]

G. LaiW. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523.  doi: 10.3934/dcds.2011.31.489.  Google Scholar

[8]

J. Q. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.  doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar

[9] T. T. Li and T. H. Qin, Physics and Partial Differential Equations (in Chinese), Higher Education Press, 2005.   Google Scholar
[10]

Y. J. Liu and W. H. Sun, Riemann problem and wave interactions in Magnetogasdynamics, J. Math. Anal. Appl., 397 (2013), 454-466.  doi: 10.1016/j.jmaa.2012.07.064.  Google Scholar

[11]

R. Singh and L. P. Singh., Solution of the Riemann Problem in magnetogasdynamics, International Journal of Non-Linear Mechanics, 67 (2014), 326-330.  doi: 10.1016/j.ijnonlinmec.2014.10.004.  Google Scholar

[12]

T. R. Sekhar and V. D. Sharma, Riemann Problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl., 11 (2010), 619-636.  doi: 10.1016/j.nonrwa.2008.10.036.  Google Scholar

[13]

D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577.  doi: 10.1007/s00205-008-0110-z.  Google Scholar

[14]

C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Appl. Math. Lett., 24 (2011), 1124-1129.  doi: 10.1016/j.aml.2011.01.038.  Google Scholar

[15]

Z. Shao, Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish, Mathematics, 2015, arXiv: 1503.08382. Google Scholar

[16]

V. M. Shelkovich, The Riemann problem admitting δ, δ'-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500.  doi: 10.1016/j.jde.2006.08.003.  Google Scholar

[17]

W. C. ShengG. J. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Anal. Real World Appl., 22 (2015), 115-128.  doi: 10.1016/j.nonrwa.2014.08.007.  Google Scholar

[18]

W. C. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (1999), 654.  doi: 10.1090/memo/0654.  Google Scholar

[19]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci., 6 (1939), 399-407.  doi: 10.1016/B978-0-12-398277-3.50005-1.  Google Scholar

[20]

D. TanT. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.  Google Scholar

[21]

G. D. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar

[22]

G. Yin and W. C. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. Appl., 12 (2009), 1-12.  doi: 10.1016/j.jmaa.2009.01.075.  Google Scholar

Figure 1.  The elementary wave curves in the phase plane
Figure 2.  The curves of elementary waves
Figure 3.  The limiting behaviors of the curves of elementary waves
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