zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Riemann problem for the relativistic Chaplygin Euler equations. (English) Zbl 1220.35126
Summary: The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.

MSC:
 35Q31 Euler equations 76Y05 Quantum hydrodynamics; relativistic hydrodynamics
Full Text:
References:
 [1] Taub, A. H.: Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions of perfect fluids, Phys. rev. 107, 884-900 (1957) · Zbl 0080.18306 [2] Thompson, Kevin W.: The special relativistic shock tube, J. fluid mech. 171, 365-375 (1986) · Zbl 0609.76133 · doi:10.1017/S0022112086001489 [3] Thorne, K. S.: Relativistic shocks: the taub adiabatic, Astrophys. J. 179, 897-907 (1973) [4] Weinberg, S.: Gravitation and cosmology: principles and applications of the general theory of relativity, (1972) [5] Chen, Jing: Conservation laws for the relativistic p-system, Comm. partial differential equations 20, 1605-1646 (1995) · Zbl 0877.35096 · doi:10.1080/03605309508821145 [6] Chen, Guiqiang; Li, Yachun: Stability of Riemann solutions with large oscillation for the relativistic Euler equations, J. differential equations 202, 332-353 (2004) · Zbl 1068.35173 · doi:10.1016/j.jde.2004.02.009 [7] Li, Yachun; Feng, Dongmei; Wang, Zejun: Global entropy solutions to the relativistic Euler equations for a class of large initial data, Z. angew. Math. phys. 56, 239-253 (2005) · Zbl 1067.35133 · doi:10.1007/s00033-005-4118-2 [8] Chen, Guiqiang; Lefloch, P. G.: Existence theory for the isentropic Euler equations, Arch. ration. Mech. anal. 166, 81-98 (2003) · Zbl 1027.76043 · doi:10.1007/s00205-002-0229-2 [9] Chang, T.; Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics, Pitman monogr. Surveys pure appl. Math. 41 (1989) · Zbl 0698.76078 [10] Chaplygin, S.: On gas jets, Sci. mem. Moscow univ. Math. phys. 21, 1-121 (1904) [11] Tsien, H. S.: Two dimensional subsonic flow of compressible fluids, J. aeron. Sci. 6, 399-407 (1939) · Zbl 0063.07865 [12] Von Karman, T.: Compressibility effects in aerodynamics, J. aeron. Sci. 8, 337-365 (1941) · Zbl 67.0853.01 [13] Bilic, N.; Tupper, G. B.; Viollier, R. D.: Dark matter, dark energy and the Chaplygin gas · Zbl 1117.83394 [14] Cruz, Norman; Lepe, Samuel; Penä, Francisco: Dissipative generalized Chaplygin gas as phantom dark energy physics, Phys. lett. B 646, 177-182 (2007) [15] Gorini, V.; Kamenshchik, A.; Moschella, U.; Pasquier, V.: The Chaplygin gas as a model for dark energy · Zbl 1179.83045 [16] Setare, M. R.: Holographic Chaplygin gas model, Phys. lett. B 648, 329-332 (2007) · Zbl 1248.83179 [17] Setare, M. R.: Interacting holographic generalized Chaplygin gas model, Phys. lett. B 654, 1-6 (2007) · Zbl 1248.83179 [18] Brenier, Y.: Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. math. Fluid mech. 7, S326-S331 (2005) · Zbl 1085.35097 · doi:10.1007/s00021-005-0162-x [19] Guo, Lihui; Sheng, Wancheng; Zhang, Tong: The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. pure appl. Anal. 9, 431-458 (2010) · Zbl 1197.35164 · doi:10.3934/cpaa.2010.9.431 [20] D.J. Korchinski, Solution of a Riemann problem for a 2{$\times$}2 system of conservation laws possessing no classical weak solution, thesis, Adelphi University, 1977. [21] Tan, Dechun; Zhang, Tong: Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (I) four-J cases, J. differential equations 111, 203-254 (1994) · Zbl 0803.35085 · doi:10.1006/jdeq.1994.1081 [22] Tan, Dechun; Zhang, Tong; Zheng, Yuxi: Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. differential equations 112, No. 1, 1-32 (1994) · Zbl 0804.35077 · doi:10.1006/jdeq.1994.1093 [23] Hayes, B. T.; Lefloch, P. G.: Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9, 1547-1563 (1996) · Zbl 0908.35075 · doi:10.1088/0951-7715/9/6/009 [24] Keyfitz, B. L.; Kranzer, H. C.: Spaces of weighted measures for conservation laws with singular shock solutions, J. differential equations 118, 420-451 (1995) · Zbl 0821.35096 · doi:10.1006/jdeq.1995.1080 [25] Sheng, Wancheng; Zhang, Tong: The Riemann problem for transportation equation in gas dynamics, Mem. amer. Math. soc. 137, No. 654 (1999) · Zbl 0913.35082 [26] Li, Jiequan; Yang, Shuli; Zhang, Tong: The two-dimensional Riemann problem in gas dynamics, (1998) · Zbl 0935.76002 [27] Yang, Hanchun: Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. differential equations 159, 447-484 (1999) · Zbl 0948.35079 · doi:10.1006/jdeq.1999.3629 [28] Yang, Hanchun: Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. math. Anal. appl. 260, 18-35 (2001) · Zbl 0985.35044 · doi:10.1006/jmaa.2000.7426 [29] Cheng, Hongjun; Liu, Wanli; Yang, Hanchun: Two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states, J. math. Anal. appl. 343, 127-140 (2008) · Zbl 1139.35073 · doi:10.1016/j.jmaa.2008.01.042 [30] Danilov, V. G.; Shelkovich, V. M.: Dynamics of propagation and interaction of ${\delta}$-shock waves in conservation laws systems, J. differential equations 221, 333-381 (2005) · Zbl 1072.35121 · doi:10.1016/j.jde.2004.12.011 [31] Panov, E. Yu.; Shelkovich, V. M.: {$\delta$}’-shock waves as a new type of solutions to systems of conservation laws, J. differential equations 228, 49-86 (2006) · Zbl 1108.35116 · doi:10.1016/j.jde.2006.04.004 [32] Lax, P. D.: Hyperbolic system of conservation laws II, Comm. pure appl. Math. 10, 537-566 (1957) · Zbl 0081.08803