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

Lihui Guo, E-mail address: lihguo@126.com

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 & Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014
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. CunhaJ. 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. DevJ. 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. YuG. Rykov and 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. GuoW. 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. KongQ. 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. LaiW. 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. ShengG. 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. 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. 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. CunhaJ. 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. DevJ. 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. YuG. Rykov and 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. GuoW. 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. KongQ. 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. LaiW. 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. ShengG. 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. 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. 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.

Figure 3.1.  Riemann solution in the phase plane
Figure 4.1.  Riemann solution when $(u_+, \rho_+)\in \rm{Ⅳ}(u_-, \rho_-).$
Figure 4.2.  Riemann solution when $(u_+, \rho_+)\in \rm{Ⅰ}(u_-, \rho_-).$
Figure 4.3.  Riemann solution when $(u_+, \rho_+)\in \rm{Ⅲ}(u_-, \rho_-).$
Figure 4.4.  Riemann solution when $(u_+, \rho_+)\in \rm{Ⅱ}(u_-, \rho_-).$
[1]

Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041

[2]

Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008

[3]

Geng Lai, Wancheng Sheng, Yuxi Zheng. Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 489-523. doi: 10.3934/dcds.2011.31.489

[4]

Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431

[5]

Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661

[6]

Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733

[7]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131

[8]

Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190

[9]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471

[10]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002

[11]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[12]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47

[13]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785

[14]

Weixia Zhao. The expansion of gas from a wedge with small angle into a vacuum. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2319-2330. doi: 10.3934/cpaa.2013.12.2319

[15]

Syed M. Assad, Chjan C. Lim. Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 1-14. doi: 10.3934/dcdsb.2005.5.1

[16]

Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867

[17]

Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic & Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255

[18]

Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127

[19]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323

[20]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (161)
  • Cited by (3)

Other articles
by authors

[Back to Top]