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August  2017, 37(8): 4439-4460. doi: 10.3934/dcds.2017190

On the uniqueness of solution to generalized Chaplygin gas

Marko Nedeljkov , and  Sanja Ružičić

Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia

* Corresponding author: Marko Nedeljkov

Received  June 2016 Revised  May 2017 Published  April 2017

Fund Project: The first author is partially supported by the projects OI174024 and III44006, Serbian Ministry of Science and by the Project 114-451-2098, APV Secretariat for Science.

The main object of the paper is finding a unique solution to Riemann problem for generalized Chaplygin gas model. That is a model of the dark energy in Universe introduced in the last decade. It permits an infinite mass concentration so one has to consider solutions containing the Dirac delta function. Although it was easy to construct solution to any Riemann problem, the usual admissibility conditions, overcompressiveness, do not exclude unwanted delta-type waves when a classical solution exists. We are using Shadow Wave approach in order to solve that uniqueness problem since they are well adopted for using Lax entropy–entropy flux conditions and there is a rich family of convex entropies.

Keywords: Generalized Chaplygin model, shadow waves, entropy solutions, delta shocks, Bessel functions.
Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 76N10.
Citation: 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
References:
[1]

A. Baricz, Bounds for modified Bessel functions of the first and second kinds, Proceedings of the Edinburgh Mathematical Society, 53 (2010), 575-599.  doi: 10.1017/S0013091508001016.  Google Scholar

[2]

M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas. accelerated expansion and dark energy-matter unification, Phys. Rev., D66 (2002), 043507.  doi: 10.1103/PhysRevD.66.043507.  Google Scholar

[3]

Y. Brenier, Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations, Journal of Mathematical Fluid Mechanics, 7 (2005), 326-331.  doi: 10.1007/s00021-005-0162-x.  Google Scholar

[4] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, New York, 2000.   Google Scholar
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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.  doi: 10.1137/S0036141001399350.  Google Scholar

[6]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[7]

W. E, Y. 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.  doi: 10.1007/BF02101897.  Google Scholar

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M. E. H. Ismail, Complete monotonicity of modified bessel functions, Proceedings of the American Mathematical Society, ., 108 (1990), 353-361.  doi: 10.1090/S0002-9939-1990-0993753-9.  Google Scholar

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A. Kamenshchik, U. Moschella and V. Pasquier, An alternative to quintessence, Phys. Lett., 511 (2001), 265-268.  doi: 10.1016/S0370-2693(01)00571-8.  Google Scholar

[10]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Diff. Eq., 118 (1995), 420-451.  doi: 10.1006/jdeq.1995.1080.  Google Scholar

[11]

A. Laforgia and P. Natalini, Some inequalities for modified Bessel functions, Journal of Inequalities and Applications, 2010 (2010), Article ID 253035, 10 pages. doi: 10.1155/2010/253035.  Google Scholar

[12]

P. LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, in: IMA Volumes in Math. and its Appl. , B. L. Keyfitz, M. Shearer (EDS), Nonlinear evolution equations that change type, Springer Verlag, Vol 27,1990,126-138. doi: 10.1007/978-1-4613-9049-7_10.  Google Scholar

[13]

D. Mitrović and M. Nedeljkov, Delta shock waves as a limit of shock waves, J. Hyp. Diff. Equ., 4 (2007), 629-653.  doi: 10.1142/S021989160700129X.  Google Scholar

[14]

M. Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math., 66 (2008), 281-302.  doi: 10.1090/S0033-569X-08-01109-5.  Google Scholar

[15]

M. Nedeljkov, Shadow waves, entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal., 197 (2010), 489-537.  doi: 10.1007/s00205-009-0281-2.  Google Scholar

[16]

M. Nedeljkov, Singular shock interactions in Chaplygin gas dynamic system, J. Differ. Equations, 256 (2014), 3859-3887.  doi: 10.1016/j.jde.2014.03.002.  Google Scholar

[17]

E. Yu. Panov and V. M. Shelkovich, δ0-Shock waves as a new type of solutions to systems of conservation laws, J. Differ. Equations, 228 (2006), 49-86.  doi: 10.1016/j.jde.2006.04.004.  Google Scholar

[18] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC-Press, Boca Raton, 1995.   Google Scholar
[19]

D. Serre, Systems of Conservation Laws I, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374.  Google Scholar

[20]

M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Commun. Nonlinear Sci. Numer. Simulat., 36 (2016), 342-353.  doi: 10.1016/j.cnsns.2015.12.013.  Google Scholar

[21]

G. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar

[22]

G. N. Watson, A Treatise on The Theory of Bessel Functions, Cambridge University Press, 1966.  Google Scholar

[23]

H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differ. Equations, 252 (2012), 5951-5993.  doi: 10.1016/j.jde.2012.02.015.  Google Scholar

show all references

References:
[1]

A. Baricz, Bounds for modified Bessel functions of the first and second kinds, Proceedings of the Edinburgh Mathematical Society, 53 (2010), 575-599.  doi: 10.1017/S0013091508001016.  Google Scholar

[2]

M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas. accelerated expansion and dark energy-matter unification, Phys. Rev., D66 (2002), 043507.  doi: 10.1103/PhysRevD.66.043507.  Google Scholar

[3]

Y. Brenier, Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations, Journal of Mathematical Fluid Mechanics, 7 (2005), 326-331.  doi: 10.1007/s00021-005-0162-x.  Google Scholar

[4] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, New York, 2000.   Google Scholar
[5]

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.  doi: 10.1137/S0036141001399350.  Google Scholar

[6]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[7]

W. E, Y. 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.  doi: 10.1007/BF02101897.  Google Scholar

[8]

M. E. H. Ismail, Complete monotonicity of modified bessel functions, Proceedings of the American Mathematical Society, ., 108 (1990), 353-361.  doi: 10.1090/S0002-9939-1990-0993753-9.  Google Scholar

[9]

A. Kamenshchik, U. Moschella and V. Pasquier, An alternative to quintessence, Phys. Lett., 511 (2001), 265-268.  doi: 10.1016/S0370-2693(01)00571-8.  Google Scholar

[10]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Diff. Eq., 118 (1995), 420-451.  doi: 10.1006/jdeq.1995.1080.  Google Scholar

[11]

A. Laforgia and P. Natalini, Some inequalities for modified Bessel functions, Journal of Inequalities and Applications, 2010 (2010), Article ID 253035, 10 pages. doi: 10.1155/2010/253035.  Google Scholar

[12]

P. LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, in: IMA Volumes in Math. and its Appl. , B. L. Keyfitz, M. Shearer (EDS), Nonlinear evolution equations that change type, Springer Verlag, Vol 27,1990,126-138. doi: 10.1007/978-1-4613-9049-7_10.  Google Scholar

[13]

D. Mitrović and M. Nedeljkov, Delta shock waves as a limit of shock waves, J. Hyp. Diff. Equ., 4 (2007), 629-653.  doi: 10.1142/S021989160700129X.  Google Scholar

[14]

M. Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math., 66 (2008), 281-302.  doi: 10.1090/S0033-569X-08-01109-5.  Google Scholar

[15]

M. Nedeljkov, Shadow waves, entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal., 197 (2010), 489-537.  doi: 10.1007/s00205-009-0281-2.  Google Scholar

[16]

M. Nedeljkov, Singular shock interactions in Chaplygin gas dynamic system, J. Differ. Equations, 256 (2014), 3859-3887.  doi: 10.1016/j.jde.2014.03.002.  Google Scholar

[17]

E. Yu. Panov and V. M. Shelkovich, δ0-Shock waves as a new type of solutions to systems of conservation laws, J. Differ. Equations, 228 (2006), 49-86.  doi: 10.1016/j.jde.2006.04.004.  Google Scholar

[18] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC-Press, Boca Raton, 1995.   Google Scholar
[19]

D. Serre, Systems of Conservation Laws I, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374.  Google Scholar

[20]

M. Sun, The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Commun. Nonlinear Sci. Numer. Simulat., 36 (2016), 342-353.  doi: 10.1016/j.cnsns.2015.12.013.  Google Scholar

[21]

G. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar

[22]

G. N. Watson, A Treatise on The Theory of Bessel Functions, Cambridge University Press, 1966.  Google Scholar

[23]

H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differ. Equations, 252 (2012), 5951-5993.  doi: 10.1016/j.jde.2012.02.015.  Google Scholar

Figure 1.  Classical waves
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Figure 2.  Overcompressive SDW vs. S1+S2
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Figure 3.  Energy entropy condition
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Figure 4.  Inequalities (15) and (16)
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Figure 5.  Inequalities (15) and (16) are not enough to prove non-positivity of $\hat{E}_{\lambda}^1$
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Figure 6.  Entropies at $\Gamma_{ss}$ curve – the first entropy pair
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Figure 7.  Entropies at $\Gamma_{ss}$ curve – the second entropy pair
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