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 and 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. 

[2]

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

[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. 

[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. 

[5]

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

[6]

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

[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. 

[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. 

[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. 

[10]

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

[11]

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

[12]

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

[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.

[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. 

[15]

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

[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. 

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[25]

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

[26]

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

[27]

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

[28]

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

[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. 

[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. 

[31]

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

[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. 

[33]

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

[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. 

[35]

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

[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. 

[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. 

[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. 

show all references

References:
[1]

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

[2]

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

[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. 

[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. 

[5]

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

[6]

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

[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. 

[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. 

[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. 

[10]

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

[11]

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

[12]

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

[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.

[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. 

[15]

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

[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. 

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[25]

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

[26]

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

[27]

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

[28]

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

[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. 

[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. 

[31]

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

[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. 

[33]

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

[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. 

[35]

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

[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. 

[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. 

[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. 

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.
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