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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 958120, 10 pages
Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type
School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, China
Received 6 December 2012; Revised 1 March 2013; Accepted 18 March 2013
Academic Editor: Dongho Chae
Copyright © 2013 Hongjun Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- B. L. Keyfitz and H. C. Kranzer, “A system of nonstrictly hyperbolic conservation laws arising in elasticity theory,” Archive for Rational Mechanics and Analysis, vol. 72, no. 3, pp. 219–241, 1980.
- Y. Lu, “Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type,” Journal of Functional Analysis, vol. 261, no. 10, pp. 2797–2815, 2011.
- A. Aw and M. Rascle, “Resurrection of “second order” models of traffic flow,” SIAM Journal on Applied Mathematics, vol. 60, no. 3, pp. 916–938, 2000.
- F. Bouchut, “On zero pressure gas dynamics,” in Advances in Kinetic Theory and Computing: Selected Papers, B. Perthame, Ed., vol. 22, pp. 171–190, World Scientific, Singapore, 1994.
- G.-Q. Chen and H. Liu, “Formation of -shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,” SIAM Journal on Mathematical Analysis, vol. 34, no. 4, pp. 925–938, 2003.
- G.-Q. Chen and H. Liu, “Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,” Physica D, vol. 189, no. 1-2, pp. 141–165, 2004.
- H. Cheng, H. Yang, and Y. Zhang, “Riemann problem for the Chaplygin Euler equations of compressible fluid flow,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 11, pp. 985–992, 2010.
- H. Cheng and H. Yang, “Riemann problem for the relativistic Chaplygin Euler equations,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 17–26, 2011.
- H. Cheng and H. Yang, “Delta shock waves in chromatography equations,” Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 475–485, 2011.
- H. Cheng and H. Yang, “Delta shock waves as limits of vanishing viscosity for 2-D steady pressureless isentropic flow,” Acta Applicandae Mathematicae, vol. 113, no. 3, pp. 323–348, 2011.
- V. G. Danilov and D. Mitrovic, “Delta shock wave formation in the case of triangular hyperbolic system of conservation laws,” Journal of Differential Equations, vol. 245, no. 12, pp. 3704–3734, 2008.
- V. G. Danilov and V. M. Shelkovich, “Dynamics of propagation and interaction of -shock waves in conservation law systems,” Journal of Differential Equations, vol. 211, no. 2, pp. 333–381, 2005.
- L. Guo, W. Sheng, and T. Zhang, “The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,” Communications on Pure and Applied Analysis, vol. 9, no. 2, pp. 431–458, 2010.
- K. T. Joseph, “A Riemann problem whose viscosity solutions contain -measures,” Asymptotic Analysis, vol. 7, no. 2, pp. 105–120, 1993.
- B. L. Keyfitz and H. C. Kranzer, “Spaces of weighted measures for conservation laws with singular shock solutions,” Journal of Differential Equations, vol. 118, no. 2, pp. 420–451, 1995.
- J. Li and T. Zhang, “Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations,” in NonlInear PDE and Related Areas, G. Q. Chen, Ed., pp. 219–232, World Scientific, Singapore, 1998.
- E. Yu. Panov and V. M. Shelkovich, “-shock waves as a new type of solutions to systems of conservation laws,” Journal of Differential Equations, vol. 228, no. 1, pp. 49–86, 2006.
- V. M. Shelkovich, “The Riemann problem admitting -, -shocks, and vacuum states (the vanishing viscosity approach),” Journal of Differential Equations, vol. 231, no. 2, pp. 459–500, 2006.
- W. Sheng and T. Zhang, “The Riemann problem for the transportation equations in gas dynamics,” Memoirs of the American Mathematical Society, vol. 137, no. 654, 1999.
- D. C. Tan and T. Zhang, “Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. I. Four- cases,” Journal of Differential Equations, vol. 111, no. 2, pp. 203–254, 1994.
- D. Tan, T. Zhang, and Y. Zheng, “Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws,” Journal of Differential Equations, vol. 112, no. 1, pp. 1–32, 2004.
- H. Yang, “Riemann problems for a class of coupled hyperbolic systems of conservation laws,” Journal of Differential Equations, vol. 159, no. 2, pp. 447–484, 1999.
- H. Yang and Y. Zhang, “New developments of delta shock waves and its applications in systems of conservation laws,” Journal of Differential Equations, vol. 252, no. 11, pp. 5951–5993, 2012.
- Y. Zheng, “Systems of conservation laws with incomplete sets of eigenvectors everywhere,” in Advances in Nonlinear Partial Differential Equations and Related Areas, G. Q. Chen, Ed., pp. 399–426, Word Scientific, Singapore, 1998.
- H. Nessyahu and E. Tadmor, “Non-oscillatory central differencing for hyperbolic conservation laws,” Journal of Computational Physics, vol. 87, no. 2, pp. 408–463, 1990.