March  2018, 17(2): 391-411. doi: 10.3934/cpaa.2018022

The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system

1. 

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

3. 

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

* Corresponding author: Chun Shen.

Received  June 2017 Revised  August 2017 Published  March 2018

Fund Project: Chun Shen is supported by NSFC (11441002) and Shandong Provincial Natural Science Foundation (ZR2014AM024), Wancheng Sheng is supported by NSFC (11371240) and Meina Sun is supported by NSFC (11271176).

The Riemann problem for the scaled Leroux system is considered. It is proven rigorously that the Riemann solutions for the scaled Leroux system converge to the corresponding ones for a non-strictly hyperbolic system of conservation laws when the perturbation parameter tends to zero. In addition, some interesting phenomena are displayed in the limiting process, such as the formation of delta shock wave and a rarefaction (or shock) wave degenerates to be a contact discontinuity.

Citation: Chun Shen, Wancheng Sheng, Meina Sun. The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system. Communications on Pure & Applied Analysis, 2018, 17 (2) : 391-411. doi: 10.3934/cpaa.2018022
References:
[1]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.   Google Scholar

[2]

E. Canon, On some hyperbolic systems of temple class, Nonlinear Anal. TMA, 75 (2012), 4241-4250.   Google Scholar

[3]

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.   Google Scholar

[4]

G. Q. Chen and H. Liu, Concentration and cavition in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141-165.   Google Scholar

[5]

Z. Cheng, On the application of kinetic formulation of the Le roux system, Proceedings of the Edinburgh Mathematical Society, 52 (2009), 263-272.   Google Scholar

[6]

G. Dal MasoP. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures. Appl., 74 (1995), 483-548.   Google Scholar

[7]

V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245 (2008), 3704-3734.   Google Scholar

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of δ-shock waves in conservation law systems, J. Differential Equations, 221 (2005), 333-381.   Google Scholar

[9]

J. Fritz and B. Toth, Derivation of Leroux system as the hydrodynamic limit of a two-component lattice gas, Comm. Math. Phys., 249 (2004), 1-27.   Google Scholar

[10]

T. Gramchev, Entropy solutions to conservation laws with singular initial data, Nonlinear Anal. TMA, 24 (1995), 721-733.   Google Scholar

[11]

F. Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations, 30 (2005), 283-304.   Google Scholar

[12]

F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146.   Google Scholar

[13]

P. Ji and C. Shen, Construction of the global solutions to the perturbed Riemann problem for the Leroux system, Advance in Mathematical Physics, 2016 (2016), 4808610, 13 pages.  Google Scholar

[14]

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.   Google Scholar

[15]

H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350.   Google Scholar

[16]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservaion laws with singular shock solutions, J. Differential Equations, 118 (1995), 420-451.   Google Scholar

[17]

A. Y. Leroux, Approximation des systems hyperboliques, in "Cours et Seminaires INRIA, problemes hyperboliques", Rocquencourt, 1981. Google Scholar

[18]

J. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.   Google Scholar

[19]

J. Li, T. Zhang and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 98, New York: Longman Scientific and Technical, 1998.  Google Scholar

[20]

Y. G. Lu, Global entropy solutions of Cauchy problem for the Le Roux system, Appl. Math. Lett., 60 (2016), 61-66.   Google Scholar

[21]

Y. G. LuI. Mantilla and L. Rendon, Convergence of approximated solutions to a nonstrictly hyperbolic system, Adv. Nonlin. Studies, 1 (2001), 65-79.   Google Scholar

[22]

D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Differential Equations, 4 (2007), 629-653.   Google Scholar

[23]

M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487-537.   Google Scholar

[24]

V. Popkov and G. M. Schutz, Why spontaneous symmetry breaking disappeas in a bridge system with PDE-friendly boundaries, J. Stat. Mech. , 12 (2004), p12004. Google Scholar

[25]

D. Serre, Solutions á variations bornées pour certains systémes hyperboliques de lois de conservation, J. Differential Equations, 68 (1987), 137-168.   Google Scholar

[26]

D. Serre, Systems of Conservation Laws 1/2, Cambridge Univ. Press, Cambridge, 1999/2000.  Google Scholar

[27]

M. Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc., 190(N889) (2007), 1-163.   Google Scholar

[28]

C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681-695.   Google Scholar

[29]

C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model, J. Differential Equations, 249 (2010), 3024-3051.   Google Scholar

[30]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137(N654) (1999), 1-77.   Google Scholar

[31]

M. Sun, Singular solutions to the Riemann problem for a macroscopic production model, Z. Angew. Math. Mech., 97 (2017), 916-931.   Google Scholar

[32]

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.   Google Scholar

[33]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.   Google Scholar

[34]

B. Toth and B. Valko, Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit, Comm. Math. Phys., 256 (2005), 111-157.   Google Scholar

[35]

A. I. Volpert, The space $BV$ and quasilinear equations, Math. USSR Sb., 2 (1967), 225-267.   Google Scholar

[36]

H. Yang and J. Wang, Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas, J. Math. Anal. Appl., 413 (2014), 800-820.   Google Scholar

[37]

H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252 (2012), 5951-5993.   Google Scholar

[38]

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. Appl., 355 (2009), 594-605.   Google Scholar

show all references

References:
[1]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.   Google Scholar

[2]

E. Canon, On some hyperbolic systems of temple class, Nonlinear Anal. TMA, 75 (2012), 4241-4250.   Google Scholar

[3]

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.   Google Scholar

[4]

G. Q. Chen and H. Liu, Concentration and cavition in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141-165.   Google Scholar

[5]

Z. Cheng, On the application of kinetic formulation of the Le roux system, Proceedings of the Edinburgh Mathematical Society, 52 (2009), 263-272.   Google Scholar

[6]

G. Dal MasoP. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures. Appl., 74 (1995), 483-548.   Google Scholar

[7]

V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245 (2008), 3704-3734.   Google Scholar

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of δ-shock waves in conservation law systems, J. Differential Equations, 221 (2005), 333-381.   Google Scholar

[9]

J. Fritz and B. Toth, Derivation of Leroux system as the hydrodynamic limit of a two-component lattice gas, Comm. Math. Phys., 249 (2004), 1-27.   Google Scholar

[10]

T. Gramchev, Entropy solutions to conservation laws with singular initial data, Nonlinear Anal. TMA, 24 (1995), 721-733.   Google Scholar

[11]

F. Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations, 30 (2005), 283-304.   Google Scholar

[12]

F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146.   Google Scholar

[13]

P. Ji and C. Shen, Construction of the global solutions to the perturbed Riemann problem for the Leroux system, Advance in Mathematical Physics, 2016 (2016), 4808610, 13 pages.  Google Scholar

[14]

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.   Google Scholar

[15]

H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350.   Google Scholar

[16]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservaion laws with singular shock solutions, J. Differential Equations, 118 (1995), 420-451.   Google Scholar

[17]

A. Y. Leroux, Approximation des systems hyperboliques, in "Cours et Seminaires INRIA, problemes hyperboliques", Rocquencourt, 1981. Google Scholar

[18]

J. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.   Google Scholar

[19]

J. Li, T. Zhang and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 98, New York: Longman Scientific and Technical, 1998.  Google Scholar

[20]

Y. G. Lu, Global entropy solutions of Cauchy problem for the Le Roux system, Appl. Math. Lett., 60 (2016), 61-66.   Google Scholar

[21]

Y. G. LuI. Mantilla and L. Rendon, Convergence of approximated solutions to a nonstrictly hyperbolic system, Adv. Nonlin. Studies, 1 (2001), 65-79.   Google Scholar

[22]

D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Differential Equations, 4 (2007), 629-653.   Google Scholar

[23]

M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487-537.   Google Scholar

[24]

V. Popkov and G. M. Schutz, Why spontaneous symmetry breaking disappeas in a bridge system with PDE-friendly boundaries, J. Stat. Mech. , 12 (2004), p12004. Google Scholar

[25]

D. Serre, Solutions á variations bornées pour certains systémes hyperboliques de lois de conservation, J. Differential Equations, 68 (1987), 137-168.   Google Scholar

[26]

D. Serre, Systems of Conservation Laws 1/2, Cambridge Univ. Press, Cambridge, 1999/2000.  Google Scholar

[27]

M. Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc., 190(N889) (2007), 1-163.   Google Scholar

[28]

C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681-695.   Google Scholar

[29]

C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model, J. Differential Equations, 249 (2010), 3024-3051.   Google Scholar

[30]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137(N654) (1999), 1-77.   Google Scholar

[31]

M. Sun, Singular solutions to the Riemann problem for a macroscopic production model, Z. Angew. Math. Mech., 97 (2017), 916-931.   Google Scholar

[32]

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.   Google Scholar

[33]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.   Google Scholar

[34]

B. Toth and B. Valko, Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit, Comm. Math. Phys., 256 (2005), 111-157.   Google Scholar

[35]

A. I. Volpert, The space $BV$ and quasilinear equations, Math. USSR Sb., 2 (1967), 225-267.   Google Scholar

[36]

H. Yang and J. Wang, Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas, J. Math. Anal. Appl., 413 (2014), 800-820.   Google Scholar

[37]

H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252 (2012), 5951-5993.   Google Scholar

[38]

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. Appl., 355 (2009), 594-605.   Google Scholar

Figure 1.  The phase plane for the scaled Leroux system (1.2) when $u_{-}<0$, left for $\varepsilon>0$ and right for the limit $\varepsilon\rightarrow0$ situation.
Figure 2.  The phase plane for the scaled Leroux system (1.2) when $u_{-}>0$, left for $\varepsilon>0$ and right for the limit $\varepsilon\rightarrow0$ situation.
[1]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[2]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[3]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[4]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[5]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[6]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[7]

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, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[8]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[9]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226

[10]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[11]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[12]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[13]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[14]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[15]

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

[16]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293

[17]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[18]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[19]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85

[20]

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

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (77)
  • HTML views (178)
  • Cited by (6)

Other articles
by authors

[Back to Top]