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The regularity of some vectorvalued variational inequalities with gradient constraints
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 
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 nonstrictly 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.
References:
[1] 
Y. Brenier, E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 23172328. 
[2] 
E. Canon, On some hyperbolic systems of temple class, Nonlinear Anal. TMA, 75 (2012), 42414250. 
[3] 
G. Q. Chen, 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), 925938. 
[4] 
G. Q. Chen, H. Liu, Concentration and cavition in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141165. 
[5] 
Z. Cheng, On the application of kinetic formulation of the Le roux system, Proceedings of the Edinburgh Mathematical Society, 52 (2009), 263272. 
[6] 
G. Dal Maso, P. G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures. Appl., 74 (1995), 483548. 
[7] 
V. G. Danilov, D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245 (2008), 37043734. 
[8] 
V. G. Danilov, V. M. Shelkovich, Dynamics of propagation and interaction of δshock waves in conservation law systems, J. Differential Equations, 221 (2005), 333381. 
[9] 
J. Fritz, B. Toth, Derivation of Leroux system as the hydrodynamic limit of a twocomponent lattice gas, Comm. Math. Phys., 249 (2004), 127. 
[10] 
T. Gramchev, Entropy solutions to conservation laws with singular initial data, Nonlinear Anal. TMA, 24 (1995), 721733. 
[11] 
F. Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations, 30 (2005), 283304. 
[12] 
F. Huang, Z. Wang, Wellposedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117146. 
[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, D. Mitrovic, Singular solutions of a fully nonlinear $2×2$ system of conservation laws, Proceedings of the Edinburgh Mathematical Society, 55 (2012), 711729. 
[15] 
H. Kalisch, D. Mitrovic, Singular solutions for the shallowwater equations, IMA J. Appl. Math., 77 (2012), 340350. 
[16] 
B. L. Keyfitz, H. C. Kranzer, Spaces of weighted measures for conservaion laws with singular shock solutions, J. Differential Equations, 118 (1995), 420451. 
[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), 519523. 
[19] 
J. Li, T. Zhang and S. Yang, The TwoDimensional 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), 6166. 
[21] 
Y. G. Lu, I. Mantilla, L. Rendon, Convergence of approximated solutions to a nonstrictly hyperbolic system, Adv. Nonlin. Studies, 1 (2001), 6579. 
[22] 
D. Mitrovic, M. Nedeljkov, Deltashock waves as a limit of shock waves, J. Hyperbolic Differential Equations, 4 (2007), 629653. 
[23] 
M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487537. 
[24] 
V. Popkov and G. M. Schutz, Why spontaneous symmetry breaking disappeas in a bridge system with PDEfriendly 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), 137168. 
[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), 1163. 
[28] 
C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681695. 
[29] 
C. Shen, M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed AwRascle model, J. Differential Equations, 249 (2010), 30243051. 
[30] 
W. Sheng, T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137(N654) (1999), 177. 
[31] 
M. Sun, Singular solutions to the Riemann problem for a macroscopic production model, Z. Angew. Math. Mech., 97 (2017), 916931. 
[32] 
D. Tan, T. Zhang, Y. Zheng, Deltashock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 132. 
[33] 
B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781795. 
[34] 
B. Toth, B. Valko, Perturbation of singular equilibria of hyperbolic twocomponent systems: a universal hydrodynamic limit, Comm. Math. Phys., 256 (2005), 111157. 
[35] 
A. I. Volpert, The space $BV$ and quasilinear equations, Math. USSR Sb., 2 (1967), 225267. 
[36] 
H. Yang, J. Wang, Deltashocks 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), 800820. 
[37] 
H. Yang, Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252 (2012), 59515993. 
[38] 
G. Yin, 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), 594605. 
show all references
References:
[1] 
Y. Brenier, E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 23172328. 
[2] 
E. Canon, On some hyperbolic systems of temple class, Nonlinear Anal. TMA, 75 (2012), 42414250. 
[3] 
G. Q. Chen, 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), 925938. 
[4] 
G. Q. Chen, H. Liu, Concentration and cavition in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141165. 
[5] 
Z. Cheng, On the application of kinetic formulation of the Le roux system, Proceedings of the Edinburgh Mathematical Society, 52 (2009), 263272. 
[6] 
G. Dal Maso, P. G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures. Appl., 74 (1995), 483548. 
[7] 
V. G. Danilov, D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245 (2008), 37043734. 
[8] 
V. G. Danilov, V. M. Shelkovich, Dynamics of propagation and interaction of δshock waves in conservation law systems, J. Differential Equations, 221 (2005), 333381. 
[9] 
J. Fritz, B. Toth, Derivation of Leroux system as the hydrodynamic limit of a twocomponent lattice gas, Comm. Math. Phys., 249 (2004), 127. 
[10] 
T. Gramchev, Entropy solutions to conservation laws with singular initial data, Nonlinear Anal. TMA, 24 (1995), 721733. 
[11] 
F. Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations, 30 (2005), 283304. 
[12] 
F. Huang, Z. Wang, Wellposedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117146. 
[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, D. Mitrovic, Singular solutions of a fully nonlinear $2×2$ system of conservation laws, Proceedings of the Edinburgh Mathematical Society, 55 (2012), 711729. 
[15] 
H. Kalisch, D. Mitrovic, Singular solutions for the shallowwater equations, IMA J. Appl. Math., 77 (2012), 340350. 
[16] 
B. L. Keyfitz, H. C. Kranzer, Spaces of weighted measures for conservaion laws with singular shock solutions, J. Differential Equations, 118 (1995), 420451. 
[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), 519523. 
[19] 
J. Li, T. Zhang and S. Yang, The TwoDimensional 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), 6166. 
[21] 
Y. G. Lu, I. Mantilla, L. Rendon, Convergence of approximated solutions to a nonstrictly hyperbolic system, Adv. Nonlin. Studies, 1 (2001), 6579. 
[22] 
D. Mitrovic, M. Nedeljkov, Deltashock waves as a limit of shock waves, J. Hyperbolic Differential Equations, 4 (2007), 629653. 
[23] 
M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487537. 
[24] 
V. Popkov and G. M. Schutz, Why spontaneous symmetry breaking disappeas in a bridge system with PDEfriendly 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), 137168. 
[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), 1163. 
[28] 
C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681695. 
[29] 
C. Shen, M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed AwRascle model, J. Differential Equations, 249 (2010), 30243051. 
[30] 
W. Sheng, T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137(N654) (1999), 177. 
[31] 
M. Sun, Singular solutions to the Riemann problem for a macroscopic production model, Z. Angew. Math. Mech., 97 (2017), 916931. 
[32] 
D. Tan, T. Zhang, Y. Zheng, Deltashock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 132. 
[33] 
B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781795. 
[34] 
B. Toth, B. Valko, Perturbation of singular equilibria of hyperbolic twocomponent systems: a universal hydrodynamic limit, Comm. Math. Phys., 256 (2005), 111157. 
[35] 
A. I. Volpert, The space $BV$ and quasilinear equations, Math. USSR Sb., 2 (1967), 225267. 
[36] 
H. Yang, J. Wang, Deltashocks 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), 800820. 
[37] 
H. Yang, Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252 (2012), 59515993. 
[38] 
G. Yin, 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), 594605. 
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