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March  2013, 12(2): 755-769. doi: 10.3934/cpaa.2013.12.755

The Riemann problem of conservation laws in magnetogasdynamics

Yanbo Hu 1, and  Wancheng Sheng 1,

1. 

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

Received  July 2011 Revised  July 2012 Published  September 2012

In this paper, we study the Riemann problem for a simplified model of one dimensional ideal gas in magnetogasdynamics. By using the characteristic analysis method, we prove the global existence of solutions to the Riemann problem constructively under the Lax entropy condition. The image of contact discontinuity in magnetogasdynamics is a curve with $u=Const.$ in the $(\tau,p,u)$ space. Its projection on the $(p,u)$ plane is a straight line that parallels to the $p$-axis. In contrast with the problem in gas dynamics, the result causes more complicated and difficult than that in gas dynamics.
Keywords: Riemann problem, Magnetogasdynamics, rarefaction wave, shock wave, contact discontinuity..
Mathematics Subject Classification: 35L65, 35L67, 76L05, 76N1.
Citation: Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
References:
[1]

W. R. Hu, "Universe Magnetogasdynamics" (in Chinese), Science Press, Beijing, 1987. Google Scholar

[2]

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

[3]

H. Cabannes, "Theoretical Magnetofluid Dynamics, in: Applid Mathematics and Mechanics," Academic Press, New York, 1970. Google Scholar

[4]

R. Gundersen, "Linearized Analysis of One-dimensional Magnetohydrodynamic Flows," Springer-Verlag, Berlin, 1964.  Google Scholar

[5]

R. S. Myong and P. L. Roe, Shock waves and rarefaction waves in magnetohydrodynamics part 1. a model system, J. Plasma Phys., 58 (1997), 485-519. doi: 10.1017/S002237789700593X.  Google Scholar

[6]

R. S. Myong and P. L. Roe, Shock waves and rarefaction waves in magnetohydrodynamics part 2. the MHD system, J. Plasma Phys., 58 (1997), 521-552. doi: 10.1017/S0022377897005941.  Google Scholar

[7]

M. Torrilhon, "Exact Solver and Uniqueness Conditions for Riemann Problem of Ideal Magnetohydrodynamics," Seminar für Angewandte Mathematik Eidgenössische Technische Hochschule, Switzerland, 2002. Google Scholar

[8]

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

[9]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Intersience Publisher, New York, 1999.  Google Scholar

[10]

P. D. Lax, Hyperbolic system of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[11]

J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm. Pure Appl. Math., 18 (1963), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[12]

T. P. Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc., 212 (1975), 375-382. doi: 10.1090/S0002-9947-1975-0380135-2.  Google Scholar

[13]

T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Longman, Harlow, 1989.  Google Scholar

[14]

J. Smollor, "Shock Waves and Reaction Diffusion Equations," Springer Verlag, New York, 1994.  Google Scholar

[15]

Y. B. Hu and W. C. Sheng, Elementary waves of conservation laws in magnetogasdynamics (in Chinese), Commun. Appl. Math. Comput., 23 (2009), 49-54.  Google Scholar

show all references

References:
[1]

W. R. Hu, "Universe Magnetogasdynamics" (in Chinese), Science Press, Beijing, 1987. Google Scholar

[2]

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

[3]

H. Cabannes, "Theoretical Magnetofluid Dynamics, in: Applid Mathematics and Mechanics," Academic Press, New York, 1970. Google Scholar

[4]

R. Gundersen, "Linearized Analysis of One-dimensional Magnetohydrodynamic Flows," Springer-Verlag, Berlin, 1964.  Google Scholar

[5]

R. S. Myong and P. L. Roe, Shock waves and rarefaction waves in magnetohydrodynamics part 1. a model system, J. Plasma Phys., 58 (1997), 485-519. doi: 10.1017/S002237789700593X.  Google Scholar

[6]

R. S. Myong and P. L. Roe, Shock waves and rarefaction waves in magnetohydrodynamics part 2. the MHD system, J. Plasma Phys., 58 (1997), 521-552. doi: 10.1017/S0022377897005941.  Google Scholar

[7]

M. Torrilhon, "Exact Solver and Uniqueness Conditions for Riemann Problem of Ideal Magnetohydrodynamics," Seminar für Angewandte Mathematik Eidgenössische Technische Hochschule, Switzerland, 2002. Google Scholar

[8]

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

[9]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Intersience Publisher, New York, 1999.  Google Scholar

[10]

P. D. Lax, Hyperbolic system of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[11]

J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm. Pure Appl. Math., 18 (1963), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[12]

T. P. Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc., 212 (1975), 375-382. doi: 10.1090/S0002-9947-1975-0380135-2.  Google Scholar

[13]

T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Longman, Harlow, 1989.  Google Scholar

[14]

J. Smollor, "Shock Waves and Reaction Diffusion Equations," Springer Verlag, New York, 1994.  Google Scholar

[15]

Y. B. Hu and W. C. Sheng, Elementary waves of conservation laws in magnetogasdynamics (in Chinese), Commun. Appl. Math. Comput., 23 (2009), 49-54.  Google Scholar

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