Abstract
This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity. Since the stress function is neither convex nor concave, the shock condition is degenerate. By introducing a degenerate shock under the generalized shock condition, the global solutions are constructively obtained case by case.
Similar content being viewed by others
References
Lu, Y. Nonlinearly degenerate wave equation v tt = c(|v|s−1 v)xx. Rev. Axad. Colombiana Cienc. Exact. Fís. Natur. 119(31), 275–283 (2007)
Smoller, J. Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York (1992)
Nishihara, K. Stability of traveling waves with degenerate shock for system of one-dimensional viscoelastic model. J. Differential Equations 120, 304–318 (1995)
Hoff, D. Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations. SIAM J. Numer. Anal. 15, 1161–1177 (1978)
Huang, F. and Wang, Z. Convergence of viscosity solutions for isentropic gas dynamics. SIAM J. Math. Anal. 34, 595–610 (2003)
Kawashima, S. and Matsumura, A. Stability of shock profiles in viscoelasticity with nonconvex constitutive relations. Comm. Pure Appl. Math. 47, 1547–1569 (1994)
Lu, Y. Existence of a global solution for a viscoelastic system. J. Math. Anal. Appl. 218, 175–182 (1998)
Matsumura, A. and Nishihara, K. Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Phys. 165, 83–96 (1994)
Zhu, C. Convergence of the viscosity solutions for the system of nonlinear elasticity. J. Math. Anal. Appl. 209, 585–604 (1997)
Chang, T. and Hsiao, L. The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Scientific and Technical, Essex (1989)
Courant, R. and Friedrichs, K. Supersonic Flow and Shock Waves, Springer, New York (1948)
Lax, P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia (1973)
Yang, H. and Sun, W. The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws. Nonlinear Anal. 67, 3041–3049 (2007)
Sun, W. and Sheng, W. The non-selfsimilar Riemann problem for 2-D zero-pressure flow in gas dynamics. Chin. Ann. Math. 28B(6), 701–708 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Xing-ming GUO
Project supported by the National Natural Science Foundation of China (No. 10971130) and the Shanghai Leading Academic Dissipline Project (No. J50101)
Rights and permissions
About this article
Cite this article
Sun, Wh., Sheng, Wc. The Riemann problem for nonlinear degenerate wave equations. Appl. Math. Mech.-Engl. Ed. 31, 665–674 (2010). https://doi.org/10.1007/s10483-010-1301-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-010-1301-9