# 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 problems for a class of coupled hyperbolic systems of conservation laws. (English) Zbl 0948.35079
The Riemann problem is solved for a family of $2\times 2$ conservation laws in one space dimension and time. The system has a double linearly degenerate charactistic and constant eigenvector. The Riemann problem solutions have so-called vacuum states and delta-shocks. Such solutions are investigated using the concept of measure-valued solution, and are realized as the limits of scale invariant solutions of the equations augmented by a form of the Dafermos regularization.

##### MSC:
 35L65 Conservation laws
Full Text:
##### References:
 [1] Adames, R. A.: Sobolev space. (1975) [2] Bouchut, F.: On zero-pressure gas dynamics. Series on advances in mathematics for applied sciences 22 (1994) · Zbl 0863.76068 [3] Courant, R.; Friedrichs, K. O.: Supersonic flow and shock waves. (1948) · Zbl 0041.11302 [4] Chen, G. Q.; Frid, H.: Existence and asymptotic behavior of measure-valued solutions for degenerate conservation laws. J. differential equations 127, 197-224 (1996) · Zbl 0854.35066 [5] Chang, T.; Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics. (1989) · Zbl 0698.76078 [6] Dafermos, C. M.: Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by viscosity method. Arch. rational mech. Anal. 52, 1-9 (1973) · Zbl 0262.35034 [7] Weinan, E.; Rykov, Yu.G.; Sinai, Ya.G.: 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, 349-380 (1996) · Zbl 0852.35097 [8] Forester, A.; Le Floch, P.: Multivalued solutions to some nonlinear and nonstrictly hyperbolic systems. Japan J. Indust. appl. Math. 9, 1-23 (1992) · Zbl 0768.35058 [9] Gelfand, I.: Some problem in the theory of quasilinear equations. Uspekhi mat. Nauk 14, 87-158 (1959) [10] Hopf, E.: The partial differential equation $ut+uux={\mu}$uxx. Comm. pure appl. Math. 3, 201-230 (1950) · Zbl 0039.10403 [11] Joseph, K. T.: A Riemann problem whose viscosity solutions contain delta-measures. Asymptotic anal. 7, 105-120 (1993) · Zbl 0791.35077 [12] Korchinski, D. J.: Solutions of a Riemann problem for a $2{\times}2$ system of conservation laws possessing classical solutions. (1977) [13] Lax, P. D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. (1973) · Zbl 0268.35062 [14] Le Floch, P.: An existence and uniqueness result for two nonstrictly hyperbolic systems. IMA math. Appl. 27 (1990) · Zbl 0727.35083 [15] Liu, T. P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. amer. Math. soc. 328 (1986) [16] Li, J.; Zhang, T.: Generalized rankine--hugoniot relations of delta-shocks in solutions of transportation equations. Nonlinear PDE and related areas, 219-232 (1998) · Zbl 0929.35092 [17] Smoller, J.: Shock waves and reaction-diffusion equations. (1992) · Zbl 0508.35002 [18] Schaeffer, D.; Shearer, M.: Riemann problems for nonstrictly hyperbolic $2{\times}2$ systems of conservation laws. Trans. amer. Math. soc. 304, 267-306 (1987) · Zbl 0656.35081 [19] Sheng, W.; Zhang, T.: The Riemann problem for transportation equations in gas dynamics. Mem. amer. Math. soc. 137 (1999) · Zbl 0913.35082 [20] Tan, D.; Zhang, T.: 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 [21] Tan, D.; Zhang, T.; Zheng, Y.: Delta-shock waves as limits of vanishing viscosity for hyperbolic system of conservation laws. J. differential equations 112, 1-32 (1994) · Zbl 0804.35077 [22] A. I. Vol’pert, and, S. I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, 1985. [23] H. Yang, and, J. Li, Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics, Quarterly Appl. Math, in press. · Zbl 1019.76040 [24] Y. Zheng, Systems of conservation laws with incomplete sets of eigenvectors everywhere, preprint, 1997. [25] Zhang, T.; Zheng, Y.: Conjecture on the structure of solution of the Riemann problem of two-dimensional gas dynamics systems. SIAM J. Math. anal. 21, 593-625 (1990) · Zbl 0726.35081