# American Institute of Mathematical Sciences

January  2017, 16(1): 295-310. doi: 10.3934/cpaa.2017014

## The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term

 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China 2 Department of Mathematics, University of Iowa Iowa, City, IA 52242, USA

Received  May 2016 Revised  July 2016 Published  November 2016

Fund Project: This work is partially supported by National Natural Science Foundation of China (11401508,11461066), China Scholarship Council, the Scientific Research Program of the Higher Education Institution of XinJiang (XJEDU2014I001).

We study the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. The phenomena of concentration and cavitation to Chaplygin gas equations with a friction term are identified and analyzed as the pressure vanishes. Due to the influence of source term, the Riemann solutions are no longer self-similar. When the pressure vanishes, the Riemann solutions to the inhomogeneous Chaplygin gas equations converge to the Riemann solutions to the pressureless gas dynamics model with a friction term.

Citation: Lihui Guo, Tong Li, Gan Yin. The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Communications on Pure and Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014
##### References:
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Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177 (1996), 349-380. [13] L. Guo, T. Li, L. Pan and X. Han, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term, submitted, 2016. [14] L. Guo, W. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [15] D. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space R1+n, Communications in Mathematical physics, 269 (2007), 153-174. doi: 10.1007/s00220-006-0124-z. [16] D. Kong and C. Wei, Formation and propagation of singularities in one-dimensional Chaplygin gas, Journal of Geometry and Physics, 80 (2014), 58-70. doi: 10.1016/j.geomphys.2014.02.009. [17] G. Lai, W. Sheng and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Discrete Contin. Dynam. Systems, 31 (2011), 489-523. doi: 10.3934/dcds.2011.31.489. [18] J. Li, Note on the compressible Euler equations with zero temperature, Applied Math. Letters, 14 (2001), 519-523. doi: 10.1016/S0893-9659(00)00187-7. [19] J. Li, T. Zhang and S. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman monographs and surveys in pure and applied athematics 98, London-New York, Longman, 1998. [20] D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Diff. Eq., 4 (2007), 629-653. doi: 10.1142/S021989160700129X. [21] M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487-537. doi: 10.1007/s00205-009-0281-2. [22] M. R. Setare, Holographic Chaplygin gas model, Phys. Lett. B, 648 (2007), 329-332. [23] S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys., 61 (1989), 185-220. doi: 10.1103/RevModPhys.61.185. [24] C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Applied Mathematics Letters, 24 (2011), 1124-1129. doi: 10.1016/j.aml.2011.01.038. [25] 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. doi: 10.1016/j.jde.2010.09.004. [26] 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. [27] C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA Journal of applied Mathematics, 81 (2016), 76-99. doi: 10.1093/imamat/hxv028. [28] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Memoirs of the American Mathematical Society, 137 (1999), 654. doi: 10.1090/memo/0654. [29] W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Analysis: Real World Applications, 22 (2015), 115-128. doi: 10.1016/j.nonrwa.2014.08.007. [30] M. Sun, Delta shock waves for the chromatography equations as self-similar viscosity limits, Quart. Appl. Math., 69 (2011), 425-443. doi: 10.1090/S0033-569X-2011-01207-3. [31] M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Communications in Nonlinear Science and Numerical Simulation, 36 (2016), 342-353. doi: 10.1016/j.cnsns.2015.12.013. [32] 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. [33] Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Mathematica Scientia, 3 (2012), 825-841. doi: 10.1016/S0252-9602(12)60064-2. [34] 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. doi: 10.1016/j.jmaa.2013.12.025. [35] 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. doi: 10.1016/j.jmaa.2009.01.075.

show all references

##### References:
 [1] N. Bilic, G. B. Tupper and R. Viollier, Dark matter, dark energy and the Chaplygin gas, arXiv: astro-ph/0207423. [2] 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. [3] Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), S326-S331. doi: 10.1007/s00021-005-0162-x. [4] G. Chen and H. Liu, Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, 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, Phys. D, 189 (2004), 141-165. doi: 10.1016/j.physd.2003.09.039. [6] S. Chen and A. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178. doi: 10.1137/110838091. [7] J. V. Cunha, J. S. Alcaniz and J. A. S. Lima, Cosmological constraints on Chaplygin gas dark energy from galaxy cluster x-ray and supernova data, Physical Review D, 69 (2004), 083501. doi: 10.1016/j.aml.2016.01.004. [8] D. A. E. Daw and M. Nedeljkov, Shadow waves for pressureless gas balance laws, Applied Mathematics Letters, 57 (2016), 54-59. [9] A. Dev, J. S. Alcaniz and D. Jain, Cosmological consequences of a Chaplygin gas dark energy, Physical Review D, 67 (2003), 023515. [10] G. Faccanoni and A. Mangeney, Exact solution for granular flows, Int. J. Numer. Anal. Meth. Geomech, 37 (2012), 1408-1433. [11] V. Gorini, A. Kamenshchik, U. Moschella and V. Pasquier, The Chaplygin gas as a model for dark energy, arXiv: gr-qc/0403062. [12] 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, Commun. Math. Phys., 177 (1996), 349-380. [13] L. Guo, T. Li, L. Pan and X. Han, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term, submitted, 2016. [14] L. Guo, W. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [15] D. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space R1+n, Communications in Mathematical physics, 269 (2007), 153-174. doi: 10.1007/s00220-006-0124-z. [16] D. Kong and C. Wei, Formation and propagation of singularities in one-dimensional Chaplygin gas, Journal of Geometry and Physics, 80 (2014), 58-70. doi: 10.1016/j.geomphys.2014.02.009. [17] G. Lai, W. Sheng and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Discrete Contin. Dynam. Systems, 31 (2011), 489-523. doi: 10.3934/dcds.2011.31.489. [18] J. Li, Note on the compressible Euler equations with zero temperature, Applied Math. Letters, 14 (2001), 519-523. doi: 10.1016/S0893-9659(00)00187-7. [19] J. Li, T. Zhang and S. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman monographs and surveys in pure and applied athematics 98, London-New York, Longman, 1998. [20] D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves, J. Hyperbolic Diff. Eq., 4 (2007), 629-653. doi: 10.1142/S021989160700129X. [21] M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal., 197 (2010), 487-537. doi: 10.1007/s00205-009-0281-2. [22] M. R. Setare, Holographic Chaplygin gas model, Phys. Lett. B, 648 (2007), 329-332. [23] S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys., 61 (1989), 185-220. doi: 10.1103/RevModPhys.61.185. [24] C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Applied Mathematics Letters, 24 (2011), 1124-1129. doi: 10.1016/j.aml.2011.01.038. [25] 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. doi: 10.1016/j.jde.2010.09.004. [26] 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. [27] C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA Journal of applied Mathematics, 81 (2016), 76-99. doi: 10.1093/imamat/hxv028. [28] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Memoirs of the American Mathematical Society, 137 (1999), 654. doi: 10.1090/memo/0654. [29] W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Analysis: Real World Applications, 22 (2015), 115-128. doi: 10.1016/j.nonrwa.2014.08.007. [30] M. Sun, Delta shock waves for the chromatography equations as self-similar viscosity limits, Quart. Appl. Math., 69 (2011), 425-443. doi: 10.1090/S0033-569X-2011-01207-3. [31] M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Communications in Nonlinear Science and Numerical Simulation, 36 (2016), 342-353. doi: 10.1016/j.cnsns.2015.12.013. [32] 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. [33] Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Mathematica Scientia, 3 (2012), 825-841. doi: 10.1016/S0252-9602(12)60064-2. [34] 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. doi: 10.1016/j.jmaa.2013.12.025. [35] 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. doi: 10.1016/j.jmaa.2009.01.075.
Riemann solution in the phase plane
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅳ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅰ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅲ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅱ}(u_-, \rho_-).$
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