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May  2019, 18(3): 1523-1545. doi: 10.3934/cpaa.2019073

## Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term

 1 Department of Mathematics, Yunnan Normal University, Kunming 650500, China 2 College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

* Corresponding author

Received  February 2017 Revised  February 2018 Published  November 2018

Fund Project: This work is supported by NSF of China (11501488), Yunnan Applied Basic Research Projects (2018FD015), the Scientific Research Foundation Project of Yunnan Education Department (2018JS150), Nan Hu Young Scholar Supporting Program of XYNU

The Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term are studied. The Riemann solutions exactly include two kinds: delta-shock solutions and vacuum solutions. In order to see more clearly the influence of the source term on Riemann solutions, the generalized Rankine-Hugoniot relations of delta shock waves are derived in detail, and the position, propagation speed and strength of delta shock wave are given. It is also shown that, as the source term vanishes, the Riemann solutions converge to the corresponding ones of the homogeneous system, which is just the generalized zero-pressure flow model and contains the one-dimensional zero-pressure flow as a prototypical example. Furthermore, the generalized balance relations associated with the generalized mass and momentum transportation are established for the delta-shock solution. Finally, two typical examples are presented to illustrate the application of our results.

Citation: Yu Zhang, Yanyan Zhang. Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1523-1545. doi: 10.3934/cpaa.2019073
##### References:
 [1] F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing (Series on Advances in Mathematics for Applied Sciences), World Scientific, Singapore, 22 (1994), 171-190. [2] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [3] Y. Brenier, W. Gangbo, G. Savare and M. Westdickenberg, The sticky particle dynamics with interactions, J. Math. Pures Appl., 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013. [4] G. Chen and H. Liu, Formation of delta-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. [5] G. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141-165. doi: 10.1016/j.physd.2003.09.039. [6] 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. doi: 10.1016/j.jde.2004.12.011. [7] V. G. Danilov and V. M. Shelkovich, Delta-shock waves type solution of hyperbolic systems of conservation laws, Q. Appl. Math., 63 (2005), 401-427. doi: 10.1090/S0033-569X-05-00961-8. [8] D. A. E. Daw and M. Nedeljkov, Shadow waves for pressureless gas balance laws, Appl. Math. Lett., 57 (2016), 54-59. doi: 10.1016/j.aml.2016.01.004. [9] Y. Ding and F. Huang, On a nonhomogeneous system of pressureless flow, Q. Appl. Math., 62 (2004), 509-528. doi: 10.1090/qam/2086043. [10] C. M. Edwards, S. D. Howison, H. Ockendon and J. R. Ockendon, Non-classical shallow water flows, IMA J. Appl. Math., 73 (2008), 137-157. doi: 10.1093/imamat/hxm064. [11] B. Engquist and O. Runborg, Multi-phase computations in geometrical optics, J. Comp. Appl. Math., 74 (1996), 175-192. doi: 10.1016/0377-0427(96)00023-4. [12] G. Faccanoni and A. Mangeney, Exact solution for granular flows, Int. J. Numer. Anal. Meth. Geomech., 37 (2012), 1408-1433. [13] I. Gallagher and L. Saint-Raymond, On pressureless gases driven by a strong inhomogeneous magnetic field, SIAM J. Math. Anal., 36 (2006), 1159-1176. doi: 10.1137/S0036141003435540. [14] L. Guo, T. Li and G. Yin, The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term, Commun. Pure Appl. Anal., 16 (2017), 295-309. doi: 10.3934/cpaa.2017014. [15] L. Guo, T. Li and G. Yin, The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term, J. Math. Anal. Appl., 455 (2017), 127-140. doi: 10.1016/j.jmaa.2017.05.048. [16] S. Ha, F. Huang and Y. Wang, A global unique solvability of entropic weak solution to the onedimensional pressureless Euler system with a flocking dissipation, J. Differential Equations, 257 (2014), 1333-1371. doi: 10.1016/j.jde.2014.05.007. [17] F. Huang, Weak solution to pressureless type system, Comm. Part. Diff. Eqs., 30 (2005), 283-304. doi: 10.1081/PDE-200050026. [18] F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [19] 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. [20] H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350. doi: 10.1093/imamat/hxs014. [21] B. L. Keyfitz and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, in Nonlinear Hyperbolic Problems, Springer, (1989), 185{197. doi: 10.1007/BFb0083875. [22] D. J. Korchinski, Solution of a Riemann Problem for A 2 × 2 System of Conservation Laws Possessing No Classical Weak Solution, Ph.D thesis, Adelphi University, 1977. [23] J. Li, T. Zhang and S. Yang, The two-dimensional Riemann problem in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 98, London-New York, Longman, 1998. [24] Y. Li and Y. Cao, Second order large particle difference method, Sci. China Ser. A, 8 (1985), 1024-1035 (in Chinese). [25] D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Differ. Equ., 4 (2007), 629-653. doi: 10.1142/S021989160700129X. [26] T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM. J. Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459. [27] S. B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline, Journal of Fluid Mechanics, 199 (1989), 177-215. doi: 10.1017/S0022112089000340. [28] S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys., 61 (1989), 185-220. doi: 10.1103/RevModPhys.61.185. [29] V. M. Shelkovich, The Riemann problem admitting δ, δ'-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500. doi: 10.1016/j.jde.2006.08.003. [30] C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA J. Appl. Math., 81 (2016), 76-99. doi: 10.1093/imamat/hxv028. [31] C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681-695. doi: 10.1002/zamm.201500015. [32] W. Sheng and T. Zhang, The Riemann problem for transportation equation in gas dynamics, Mem. Am. Math. Soc., 137 (1999), 1-77. doi: 10.1090/memo/0654. [33] M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 342-353. doi: 10.1016/j.cnsns.2015.12.013. [34] D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (Ⅰ) Four-J cases, Ⅱ. Initial data involving some rarefaction waves, J. Differential Equations, 111 (1994), 203-282. doi: 10.1006/jdeq.1994.1082. [35] 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. [36] H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159 (1999), 447-484. doi: 10.1006/jdeq.1999.3629. [37] H. Yang, Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260 (2001), 18-35. doi: 10.1006/jmaa.2000.7426. [38] H. Yang and W. Sun, The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Analysis Series A: Theory, Methods & Applications, 67 (2007), 3041-3049. doi: 10.1016/j.na.2006.09.057. [39] 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. doi: 10.1016/j.jde.2012.02.015. [40] H. Yang and Y. Zhang, Delta shock waves with Dirac delta function in both components for systems of conservation laws, J. Differential Equations, 257 (2014), 4369-4402. doi: 10.1016/j.jde.2014.08.009. [41] W. E Yu, G. Rykov and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. [42] Y. Zhang and Y. Zhang, Vanishing viscosity limit for Riemann solutions to a class of nonstrictly hyperbolic systems, Acta Applicandae Mathematicae, 155 (2018), 151-175. doi: 10.1007/s10440-017-0149-7.

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##### References:
 [1] F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing (Series on Advances in Mathematics for Applied Sciences), World Scientific, Singapore, 22 (1994), 171-190. [2] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [3] Y. Brenier, W. Gangbo, G. Savare and M. Westdickenberg, The sticky particle dynamics with interactions, J. Math. Pures Appl., 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013. [4] G. Chen and H. Liu, Formation of delta-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. [5] G. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Physica D, 189 (2004), 141-165. doi: 10.1016/j.physd.2003.09.039. [6] 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. doi: 10.1016/j.jde.2004.12.011. [7] V. G. Danilov and V. M. Shelkovich, Delta-shock waves type solution of hyperbolic systems of conservation laws, Q. Appl. Math., 63 (2005), 401-427. doi: 10.1090/S0033-569X-05-00961-8. [8] D. A. E. Daw and M. Nedeljkov, Shadow waves for pressureless gas balance laws, Appl. Math. Lett., 57 (2016), 54-59. doi: 10.1016/j.aml.2016.01.004. [9] Y. Ding and F. Huang, On a nonhomogeneous system of pressureless flow, Q. Appl. Math., 62 (2004), 509-528. doi: 10.1090/qam/2086043. [10] C. M. Edwards, S. D. Howison, H. Ockendon and J. R. Ockendon, Non-classical shallow water flows, IMA J. Appl. Math., 73 (2008), 137-157. doi: 10.1093/imamat/hxm064. [11] B. Engquist and O. Runborg, Multi-phase computations in geometrical optics, J. Comp. Appl. Math., 74 (1996), 175-192. doi: 10.1016/0377-0427(96)00023-4. [12] G. Faccanoni and A. Mangeney, Exact solution for granular flows, Int. J. Numer. Anal. Meth. Geomech., 37 (2012), 1408-1433. [13] I. Gallagher and L. Saint-Raymond, On pressureless gases driven by a strong inhomogeneous magnetic field, SIAM J. Math. Anal., 36 (2006), 1159-1176. doi: 10.1137/S0036141003435540. [14] L. Guo, T. Li and G. Yin, The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term, Commun. Pure Appl. Anal., 16 (2017), 295-309. doi: 10.3934/cpaa.2017014. [15] L. Guo, T. Li and G. Yin, The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term, J. Math. Anal. Appl., 455 (2017), 127-140. doi: 10.1016/j.jmaa.2017.05.048. [16] S. Ha, F. Huang and Y. Wang, A global unique solvability of entropic weak solution to the onedimensional pressureless Euler system with a flocking dissipation, J. Differential Equations, 257 (2014), 1333-1371. doi: 10.1016/j.jde.2014.05.007. [17] F. Huang, Weak solution to pressureless type system, Comm. Part. Diff. Eqs., 30 (2005), 283-304. doi: 10.1081/PDE-200050026. [18] F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [19] 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. [20] H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350. doi: 10.1093/imamat/hxs014. [21] B. L. Keyfitz and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, in Nonlinear Hyperbolic Problems, Springer, (1989), 185{197. doi: 10.1007/BFb0083875. [22] D. J. Korchinski, Solution of a Riemann Problem for A 2 × 2 System of Conservation Laws Possessing No Classical Weak Solution, Ph.D thesis, Adelphi University, 1977. [23] J. Li, T. Zhang and S. Yang, The two-dimensional Riemann problem in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 98, London-New York, Longman, 1998. [24] Y. Li and Y. Cao, Second order large particle difference method, Sci. China Ser. A, 8 (1985), 1024-1035 (in Chinese). [25] D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Differ. Equ., 4 (2007), 629-653. doi: 10.1142/S021989160700129X. [26] T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM. J. Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459. [27] S. B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline, Journal of Fluid Mechanics, 199 (1989), 177-215. doi: 10.1017/S0022112089000340. [28] S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys., 61 (1989), 185-220. doi: 10.1103/RevModPhys.61.185. [29] V. M. Shelkovich, The Riemann problem admitting δ, δ'-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459-500. doi: 10.1016/j.jde.2006.08.003. [30] C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA J. Appl. Math., 81 (2016), 76-99. doi: 10.1093/imamat/hxv028. [31] C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681-695. doi: 10.1002/zamm.201500015. [32] W. Sheng and T. Zhang, The Riemann problem for transportation equation in gas dynamics, Mem. Am. Math. Soc., 137 (1999), 1-77. doi: 10.1090/memo/0654. [33] M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 342-353. doi: 10.1016/j.cnsns.2015.12.013. [34] D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (Ⅰ) Four-J cases, Ⅱ. Initial data involving some rarefaction waves, J. Differential Equations, 111 (1994), 203-282. doi: 10.1006/jdeq.1994.1082. [35] 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. [36] H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159 (1999), 447-484. doi: 10.1006/jdeq.1999.3629. [37] H. Yang, Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260 (2001), 18-35. doi: 10.1006/jmaa.2000.7426. [38] H. Yang and W. Sun, The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Analysis Series A: Theory, Methods & Applications, 67 (2007), 3041-3049. doi: 10.1016/j.na.2006.09.057. [39] 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. doi: 10.1016/j.jde.2012.02.015. [40] H. Yang and Y. Zhang, Delta shock waves with Dirac delta function in both components for systems of conservation laws, J. Differential Equations, 257 (2014), 4369-4402. doi: 10.1016/j.jde.2014.08.009. [41] W. E Yu, G. Rykov and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. [42] Y. Zhang and Y. Zhang, Vanishing viscosity limit for Riemann solutions to a class of nonstrictly hyperbolic systems, Acta Applicandae Mathematicae, 155 (2018), 151-175. doi: 10.1007/s10440-017-0149-7.
The Riemann solution of (1) and (2) when $u_-<0<u_+$ and $\beta>0$ for a given time $t$ before the time $(f^{-1}(0)-u_-)/\beta$. The left is the $(u, v)$-phase plane, and the right is the corresponding $(x, t)$-characteristic plane
The delta-shock solution of (1) and (2) for $\beta>0$, where the propagation speed of delta shock wave is positive on the left and negative on the right when $t = 0$
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