The Riemann problem of conservation laws in magnetogasdynamics

Previous Article
Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part
CPAA Home
This Issue
Next Article
Long-time dynamics of the parabolic $p$-Laplacian equation

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

Abstract
Related Papers

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 and 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.
[2]	D. Q. Li and T. H. Qin, "Physics and Partial Differential Equations" (in Chinese), Higher Education Press, 2005.
[3]	H. Cabannes, "Theoretical Magnetofluid Dynamics, in: Applid Mathematics and Mechanics," Academic Press, New York, 1970.
[4]	R. Gundersen, "Linearized Analysis of One-dimensional Magnetohydrodynamic Flows," Springer-Verlag, Berlin, 1964.
[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.
[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.
[7]	M. Torrilhon, "Exact Solver and Uniqueness Conditions for Riemann Problem of Ideal Magnetohydrodynamics," Seminar für Angewandte Mathematik Eidgenössische Technische Hochschule, Switzerland, 2002.
[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.
[9]	R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Intersience Publisher, New York, 1999.
[10]	P. D. Lax, Hyperbolic system of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.
[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.
[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.
[13]	T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Longman, Harlow, 1989.
[14]	J. Smollor, "Shock Waves and Reaction Diffusion Equations," Springer Verlag, New York, 1994.
[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.

show all references

References:

[1]	W. R. Hu, "Universe Magnetogasdynamics" (in Chinese), Science Press, Beijing, 1987.
[2]	D. Q. Li and T. H. Qin, "Physics and Partial Differential Equations" (in Chinese), Higher Education Press, 2005.
[3]	H. Cabannes, "Theoretical Magnetofluid Dynamics, in: Applid Mathematics and Mechanics," Academic Press, New York, 1970.
[4]	R. Gundersen, "Linearized Analysis of One-dimensional Magnetohydrodynamic Flows," Springer-Verlag, Berlin, 1964.
[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.
[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.
[7]	M. Torrilhon, "Exact Solver and Uniqueness Conditions for Riemann Problem of Ideal Magnetohydrodynamics," Seminar für Angewandte Mathematik Eidgenössische Technische Hochschule, Switzerland, 2002.
[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.
[9]	R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Intersience Publisher, New York, 1999.
[10]	P. D. Lax, Hyperbolic system of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.
[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.
[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.
[13]	T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Longman, Harlow, 1989.
[14]	J. Smollor, "Shock Waves and Reaction Diffusion Equations," Springer Verlag, New York, 1994.
[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.

[1]	Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic and Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
[2]	Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271
[3]	Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
[4]	Cristóbal Rodero, J. Alberto Conejero, Ignacio García-Fernández. Shock wave formation in compliant arteries. Evolution Equations and Control Theory, 2019, 8 (1) : 221-230. doi: 10.3934/eect.2019012
[5]	Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure and Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008
[6]	Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[7]	Qingqing Liu, Xiaoli Wu. Stability of rarefaction wave for viscous vasculogenesis model. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022034
[8]	Lili Fan, Lizhi Ruan, Wei Xiang. Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1971-1999. doi: 10.3934/dcds.2020349
[9]	Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025
[10]	Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555
[11]	Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic and Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047
[12]	Anupam Sen, T. Raja Sekhar. Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2641-2653. doi: 10.3934/cpaa.2020115
[13]	Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
[14]	Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
[15]	Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure and Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
[16]	Shu Wang, Yixuan Zhao. Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022091
[17]	Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[18]	Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[19]	Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[20]	Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure and Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Yanbo Hu Wancheng Sheng
on Google Scholar
Yanbo Hu Wancheng Sheng

[Back to Top]