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August
2017, 37(8): 4439-4460.
doi: 10.3934/dcds.2017190
On the uniqueness of solution to generalized Chaplygin gas
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia |
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
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References:
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Bounds for modified Bessel functions of the first and second kinds, Proceedings of the Edinburgh Mathematical Society, 53 (2010), 575-599.
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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. |
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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.
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A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, New York, 2000.
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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. |
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C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg, 2000.
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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.
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Complete monotonicity of modified bessel functions, Proceedings of the American Mathematical Society, ., 108 (1990), 353-361.
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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. |
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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. |
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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. |
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D. Mitrović and M. Nedeljkov,
Delta shock waves as a limit of shock waves, J. Hyp. Diff. Equ., 4 (2007), 629-653.
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M. Nedeljkov,
Singular shock waves in interactions, Quart. Appl. Math., 66 (2008), 281-302.
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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. |
| [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. |
| [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. |
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A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC-Press, Boca Raton, 1995.
|
| [19] |
D. Serre, Systems of Conservation Laws I, Cambridge University Press, 1999.
doi: 10.1017/CBO9780511612374. |
| [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. |
| [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. |
| [22] |
G. N. Watson, A Treatise on The Theory of Bessel Functions, Cambridge University Press, 1966. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, New York, 2000.
|
| [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. |
| [6] |
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg, 2000.
doi: 10.1007/3-540-29089-3_14. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
M. Nedeljkov,
Singular shock waves in interactions, Quart. Appl. Math., 66 (2008), 281-302.
doi: 10.1090/S0033-569X-08-01109-5. |
| [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. |
| [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. |
| [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. |
| [18] |
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC-Press, Boca Raton, 1995.
|
| [19] |
D. Serre, Systems of Conservation Laws I, Cambridge University Press, 1999.
doi: 10.1017/CBO9780511612374. |
| [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. |
| [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. |
| [22] |
G. N. Watson, A Treatise on The Theory of Bessel Functions, Cambridge University Press, 1966. |
| [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. |
Figure 2.
Overcompressive SDW vs. S1+S2
Figure 3.
Energy entropy condition
Figure 4.
Inequalities (15) and (16)
Figure 5.
Inequalities (15) and (16) are not enough to prove non-positivity of $\hat{E}_{\lambda}^1$
Figure 6.
Entropies at $\Gamma_{ss}$ curve – the first entropy pair
Figure 7.
Entropies at $\Gamma_{ss}$ curve – the second entropy pair
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