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March  2019, 18(2): 931-942. doi: 10.3934/cpaa.2019045

## Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation

 Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

* Corresponding author

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: The first author is supported by University Grant Commission, Government of India (Sr. No. 2121540947, Ref No: 20/12/2015(ⅱ)EU-V). The second author is supported by Science and Engineering Research Board, Department of Science and Technology, Government of India (Ref No: SB/FTP/MS-047/2013).

In this article, we study the Riemann problem for a strictly hyperbolic system of conservation laws under the linear approximation of flux functions with three parameters. The approximation does not affect the structure of Riemann problem. Furthermore, we prove that the Riemann solution to the approximated system converges to the original system as the perturbation parameter tends to zero.

Citation: Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045
##### References:
 [1] G. Q. Chen and H. Liu, Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar [2] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D: Nonlinear Phenomena, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039.  Google Scholar [3] H. Cheng and H. Yang, Approaching Chaplygin pressure limit of solutions to the Aw-Rascle model, J. Math. Anal. and Appl., 416 (2014), 839-854.  doi: 10.1016/j.jmaa.2014.03.010.  Google Scholar [4] V. G. Danilov and V. M. Shelkovich, Dynamics of Propagation and interaction of δ-shock waves in conservation law systems, J. Differential Equations, 211 (2005), 333-381.  doi: 10.1016/j.jde.2004.12.011.  Google Scholar [5] V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math., 63 (2005), 401-427.  doi: 10.1090/S0033-569X-05-00961-8.  Google Scholar [6] G. Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math., 58 (2000), 177-199.  doi: 10.1090/qam/1739044.  Google Scholar [7] K. T. Joseph, A Riemann problem whose viscosity solutions contain δ-measures, Asymptotic Anal., 7 (1993), 105-120.   Google Scholar [8] H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, The IMA Journal of Applied Mathematics, 73 (2012), 340-350.  doi: 10.1093/imamat/hxs014.  Google Scholar [9] H. Kalisch and D. Mitrovic, Singular solutions of a fully nonlinear 2× 2 system of conservation laws, Proceedings of the Edinburgh Mathematical Society, 55 (2012), 711-729.  doi: 10.1017/S0013091512000065.  Google Scholar [10] E. Y. Panov and V. M. Shelkovich, δ'-Shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations, 228 (2006), 49-86.  doi: 10.1016/j.jde.2006.04.004.  Google Scholar [11] A. Sen, T. Raja Sekhar and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math., 75 (2017), 539-554.  doi: 10.1090/qam/1466.  Google Scholar [12] 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 [13] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, American Mathematical Soc., 654 (1999).  doi: 10.1090/memo/0654.  Google Scholar [14] M. Sun, Structural stability of solutions to the Riemann problem for a non-strictly hyperbolic system with flux approximation, Electronic Journal of Differential Equations, 2016 (2016), 1-16.   Google Scholar [15] D. Tan, T. 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 [16] H. Yang and J. Liu, Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation, Science China Mathematics, 58 (2015), 2329-2346.  doi: 10.1007/s11425-015-5034-0.  Google Scholar [17] G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. and Appl., 355 (2009), 594-605.  doi: 10.1016/j.jmaa.2009.01.075.  Google Scholar

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##### References:
 [1] G. Q. Chen and H. Liu, Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar [2] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D: Nonlinear Phenomena, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039.  Google Scholar [3] H. Cheng and H. Yang, Approaching Chaplygin pressure limit of solutions to the Aw-Rascle model, J. Math. Anal. and Appl., 416 (2014), 839-854.  doi: 10.1016/j.jmaa.2014.03.010.  Google Scholar [4] V. G. Danilov and V. M. Shelkovich, Dynamics of Propagation and interaction of δ-shock waves in conservation law systems, J. Differential Equations, 211 (2005), 333-381.  doi: 10.1016/j.jde.2004.12.011.  Google Scholar [5] V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math., 63 (2005), 401-427.  doi: 10.1090/S0033-569X-05-00961-8.  Google Scholar [6] G. Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math., 58 (2000), 177-199.  doi: 10.1090/qam/1739044.  Google Scholar [7] K. T. Joseph, A Riemann problem whose viscosity solutions contain δ-measures, Asymptotic Anal., 7 (1993), 105-120.   Google Scholar [8] H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, The IMA Journal of Applied Mathematics, 73 (2012), 340-350.  doi: 10.1093/imamat/hxs014.  Google Scholar [9] H. Kalisch and D. Mitrovic, Singular solutions of a fully nonlinear 2× 2 system of conservation laws, Proceedings of the Edinburgh Mathematical Society, 55 (2012), 711-729.  doi: 10.1017/S0013091512000065.  Google Scholar [10] E. Y. Panov and V. M. Shelkovich, δ'-Shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations, 228 (2006), 49-86.  doi: 10.1016/j.jde.2006.04.004.  Google Scholar [11] A. Sen, T. Raja Sekhar and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math., 75 (2017), 539-554.  doi: 10.1090/qam/1466.  Google Scholar [12] 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 [13] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, American Mathematical Soc., 654 (1999).  doi: 10.1090/memo/0654.  Google Scholar [14] M. Sun, Structural stability of solutions to the Riemann problem for a non-strictly hyperbolic system with flux approximation, Electronic Journal of Differential Equations, 2016 (2016), 1-16.   Google Scholar [15] D. Tan, T. 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 [16] H. Yang and J. Liu, Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation, Science China Mathematics, 58 (2015), 2329-2346.  doi: 10.1007/s11425-015-5034-0.  Google Scholar [17] G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. and Appl., 355 (2009), 594-605.  doi: 10.1016/j.jmaa.2009.01.075.  Google Scholar
Riemann solution of (1) and (5) in the phase plane
The Riemann solution of (4) and (5) is R+J when $u_{-}<u_+$
The Riemann solution of (4) and (5) is S+J when $u_{+}<u_{-}<u_{+}+1.$
The Riemann solution of (4) and (5) is $\delta{S}$ when $u_{-}\geq{u}_{+}+1$
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