• Previous Article
    Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations
  • CPAA Home
  • This Issue
  • Next Article
    Attractors and their stability on Boussinesq type equations with gentle dissipation
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. SenT. 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. 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

[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

show all references

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. SenT. 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. 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

[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

Figure 1.  Riemann solution of (1) and (5) in the phase plane
Figure 2.  The Riemann solution of (4) and (5) is R+J when $u_{-}<u_+$
Figure 3.  The Riemann solution of (4) and (5) is S+J when $u_{+}<u_{-}<u_{+}+1.$
Figure 4.  The Riemann solution of (4) and (5) is $\delta{S}$ when $u_{-}\geq{u}_{+}+1$
[1]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[2]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[3]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[4]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[5]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[6]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[7]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[8]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[9]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[10]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[11]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[12]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[13]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[14]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[16]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[17]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[18]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[19]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[20]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (151)
  • HTML views (188)
  • Cited by (6)

Other articles
by authors

[Back to Top]