×

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 Hyperbolic conservation laws
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adames, R. A., Sobolev Space (1975), Academic Press: Academic Press New York
[2] Bouchut, F., On zero-pressure gas dynamics, Advances in Kinetic Theory and Computing. Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, 22 (1994), World Scientific: World Scientific Singapore, p. 171-190 · Zbl 0863.76068
[3] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience: Interscience New York · 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, Pitman Monographs (1989), Longman: Longman Essex · 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) · Zbl 0096.06602
[10] Hopf, E., The partial differential equation \(u_t\)+\(uu_x\)=\(μu_{xx} \), 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 2×2 System of Conservation Laws Possessing Classical Solutions (1977), Adelphi University
[13] Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973), SIAM: SIAM Philadelphia · Zbl 0268.35062
[14] Le Floch, P., An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations That Change Type. Nonlinear Evolution Equations That Change Type, IMA Math. Appl., 27 (1990), Springer-Verlag: Springer-Verlag New York/Berlin · 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, (Chen, G. Q., Nonlinear PDE and Related Areas (1998), World Scientific: World Scientific Singapore), 219-232 · Zbl 0929.35092
[17] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1992), Springer-Verlag: Springer-Verlag New York
[18] Schaeffer, D.; Shearer, M., Riemann problems for nonstrictly hyperbolic 2×2 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.; A. I. Vol’pert, and, S. I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, 1985. · Zbl 0564.46025
[23] H. Yang, and, J. Li, Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics, Quarterly Appl. Math, in press.; 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.; 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.