The Riemann problem for the Chaplygin gas equations with a source term. (English) Zbl 1538.35221
Summary: The Riemann solutions for the one-dimensional Chaplygin gas equations with a Coulomb-like friction term are constructed explicitly. It is shown that the delta shock wave appears in the Riemann solutions in some certain situations. The generalized Rankine-Hugoniot conditions of delta shock wave are established and the position, propagation speed and strength of delta shock wave are given, which enables us to see the influence of Coulomb-like friction term on the Riemann solutions for the Chaplygin gas equations clearly. In addition, the relations connected with the area transportation are derived which include mass and momentum transportation.
{© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim}
{© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim}
MSC:
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
References:
| [1] | M. C.Bento, O.Bertolami, and A. A.Sen, Generalized Chaplygin gas, accelerated expansion and dark‐energy‐matter unification, Phys. Rev. D66(4), 043507 (2002). |
| [2] | N.Bilic, G. B.Tupper, and R. D.Viollier, Unification of dark matter and dark energy: the inhomogeneous Chaplygin gas, Physics Letters B535, 17-21 (2012). · Zbl 0995.83512 |
| [3] | Y.Brenier, Solutions with concentration to the Riemann problem for one‐dimensional Chaplygin gas equations, J. Math. Fluid Mech.7, S326-S331 (2005). · Zbl 1085.35097 |
| [4] | S.Chaplygin, On gas jets, Sci. Mem. Moscow Univ. Math. Phys.21, 1-121 (1904). |
| [5] | S.Chen and A.Qu, Two‐dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal.44, 2146-2178 (2012). · Zbl 1257.35126 |
| [6] | S.Chen and A.Qu, Riemann boundary value problems and reflection of shock for the Chaplygin gas, Sci. China A55, 671-685 (2012). · Zbl 1239.35094 |
| [7] | H.Cheng and H.Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl.381, 17-26 (2011). · Zbl 1220.35126 |
| [8] | H.Cheng and H.Yang, Riemann problem for the isentropic relativistic Chaplygin Euler equations, Z. Angew. Math. Phys.63, 429-440 (2012). · Zbl 1252.35200 |
| [9] | N.Cruz, S.Lepe, and F.Pena, Dissipative generalized Chaplygin gas as phantom dark energy, Physics Letters B646, 177-182, (2007) |
| [10] | V. G.Danilov and V. M.Shelkovich, Dynamics of propagation and interaction of δ‐shock waves in conservation law systems, J. Differential Equations221, 333-381 (2005). · Zbl 1072.35121 |
| [11] | V. G.Danilov and V. M.Shelkovich, Delta‐shock waves type solution of hyperbolic systems of conservation laws, Q. Appl. Math.63, 401-427 (2005). |
| [12] | F.Gu, Y. G.Lu, and Q.Zhang, Global solutions to one‐dimensional shallow water magnetohydrodynamic equations, J. Math. Anal. Appl.401, 714-723 (2013). · Zbl 1307.35222 |
| [13] | L.Guo, W.Sheng, and T.Zhang, The two‐dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal.9, 431-458 (2010). · Zbl 1197.35164 |
| [14] | L.Guo, Y.Zhang, and G.Yin, Interactions of delta shock waves for the Chaplygin gas equations with split delta functions, J. Math. Anal. Appl.410, 190-201 (2014). · Zbl 1312.35131 |
| [15] | L.Guo, Y.Zhang, and G.Yin, Interactions of delta shock waves for the relativistic Chaplygin gas equations with split delta functions, Math. Meth. Appl. Sci.38, 2132-2148 (2015). · Zbl 1317.35141 |
| [16] | G.Faccanoni and A.Mangeney, Exact solution for granular flows, Int. J. Numer. Anal. Meth. Geomech.37, 1408-1433 (2012). |
| [17] | F.Huang and Z.Wang, Well‐posedness for pressureless flow, Comm. Math. Phys.222, 117-146 (2001). · Zbl 0988.35112 |
| [18] | M.Jamil and M. A.Rashid, Interacting modified variable Chaplygin in a non‐flat universe, Eur. Phys. J. C58, 111-114 (2008). |
| [19] | H.Kalisch and D.Mitrovic, Singular solutions of a fully nonlinear 2 × 2 system of conservation laws, Proceedings of the Edinburgh Mathematical Society55, 711-729 (2012). · Zbl 1286.35162 |
| [20] | H.Kalisch and D.Mitrovic, Singular solutions for the shallow‐water equations, IMA J. Appl. Math.77, 340‐350 (2012). · Zbl 1257.35123 |
| [21] | K. V.Karelsky, A. S.Petrosyan, and S. V.Tarasevich, Nonlinear dynamics of magnethydrodynamic flows of heavy fluid on slope in shallow water approximation, Journal of Experimental and Theoretical Physics146, 352-367 (2014). |
| [22] | D.Kong and C.Wei, Formation and propagation of singlarities in one‐dimensional Chaplygin gas, Journal of Geometry and Physics80, 58-70 (2014). · Zbl 1284.35260 |
| [23] | D.Kong, C.Wei, and Q.Zhang, Formation of singularities in one‐dimensional Chaplygin gas, J. Hyperbolic Differential Equations11, 521-562 (2014). · Zbl 1310.35154 |
| [24] | G.Lai, W.Sheng, and Y.Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi‐dimensions, Discrete Contin. Dyn. Syst.31, 489-523 (2011). · Zbl 1222.35121 |
| [25] | J.Li, T.Zhang, and S.Yang, The Two‐Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 98 (Longman Scientific and Technical, 1998). · Zbl 0935.76002 |
| [26] | Y. G.Lu, Existence of global entropy solutions to general system of Keyfitz‐Kranzer type, J. Functional Analysis264, 2457-2468 (2013). · Zbl 1282.35240 |
| [27] | M.Nedeljkov, Higher order shadow waves and delta shock blow up in the Chaplygin gas, J. Differential Equations256, 3859-3887 (2014). · Zbl 1288.35350 |
| [28] | M.Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal.197, 487-537 (2010). · Zbl 1201.35134 |
| [29] | A.Qu and Z.Wang, Stability of the Riemann solutions for a Chaplygin gas, J. Math. Anal. Appl.409, 347-361 (2014). · Zbl 1306.35073 |
| [30] | S. B.Savage and K.Hutter, The motion of finite mass of granular material down a rough incline, Journal of Fluid Mechanics199, 177-215 (1989). · Zbl 0659.76044 |
| [31] | D.Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal.191, 539-577 (2009). · Zbl 1161.76025 |
| [32] | M. R.Setare, Holographic Chaplygin DGP cosmologies, Internat. J. Modern Phys. D18, 419-427 (2009). · Zbl 1173.83319 |
| [33] | C.Shen, The Riemann problem for the pressureless Euler system with Coulomb‐like friction term, preprint. · Zbl 1336.35281 |
| [34] | 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 Equations249, 3024-3051 (2010). · Zbl 1211.35193 |
| [35] | W.Sheng and T.Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc.137(N654), (1999), AMS: Providence. · Zbl 0913.35082 |
| [36] | J.Smoller, Shock Waves and Reaction‐Diffusion Equations (Springer‐Verlag, NewYork, 1994). · Zbl 0807.35002 |
| [37] | H. S.Tsien, Two dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci.6, 399-407 (1939). · JFM 65.1496.05 |
| [38] | T.vonKármán, Compressibility effects in aerodynamics, J. Aeronaut. Sci.8, 337-365 (1941). · JFM 67.0853.01 |
| [39] | G.Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl.403, 434-450 (2013). · Zbl 1426.76207 |
| [40] | G.Wang, B.Chen, and Y.Hu, The two‐dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl.393, 544-562 (2012). · Zbl 1250.35148 |
| [41] | Z.Wang and Q.Zhang, The Riemann problem with delta initial data for the one‐dimensional Chaplygin gas equations, Acta Math. Sci.32B, 825-841 (2012). · Zbl 1274.35232 |
| [42] | 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, 800-820 (2014). · Zbl 1308.76233 |
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