Abstract
By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely.
Similar content being viewed by others
References
Andreianov, B.P.: On viscous limit solutions of the Riemann problem for the equations of isentropic gas dynamics in Eulerian coordinates. Sb. Math. 194(6), 793–811 (2003)
Adames, R.A.: Sobolev Space. Academic Press, New York (1975)
Chen, G., Liu, H.: Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34(4), 925–938 (2003)
Chen, G., Liu, H.: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Phys. D 189(1), 141–165 (2004)
Dafermos, C.M.: Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method. Arch. Ration. Mech. Anal. 52(1), 1–9 (1973)
Dafermos, C.M., Diperna, R.J.: The Riemann problem for certain classes of hyperbolic systems of conservation laws. J. Differ. Equ. 20(1), 90–114 (1976)
Danilov, V.G., Shelkovish, V.M.: Dynamics of propagation and interaction of \(\delta \)-shock waves in conservation law systems. J. Differ. Equ. 211(2), 333–381 (2005)
Danilov, V.G., Shelkovish, V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63(3), 401–427 (2005)
Engquist, B., Runborg, O.: Multiphase computations in geometrical optics. J. Comput. Appl. Math. 74(96), 175–192 (1996)
Ercole, G.: Delta-shock waves as self-similar viscosity limits. Q. Appl. Math. LVIII(1), 177–199 (2000)
Hopf, E.: The partial differential equation \(u_{t}+uu_{x}=\mu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230 (1950)
Hu, J.: A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws. Q. Appl. Math. LV(2), 361–373 (1997)
Hu, J.: The Riemann problem for pressureless fluid dynamics with distribution solutions in Colombeau’s sense. Commun. Math. Phys. 194(1), 191–205 (1998)
Joseph, K.T.: A Riemann problem whose viscosity solutions contain \(\delta \)-measures. Asymptot. Anal. 7(2), 105–120 (1993)
Joseph, K.T.: Explicit generalized solutions to a system of conservation laws. Proc. Indian Acad. Sci. Math. Sci. 109(4), 401–409 (1999)
Kalasnikov, A.S.: Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter. Dokl. Akad. Nauk SSSR 127, 27–30 (1959)
Li, J.: Note on the compressible Euler equations with zero temperature. Appl. Math. Lett. 14(4), 519–523 (2001)
Panov, E.Y., Shelkovich, V.M.: \(\delta '\)-shock waves as a new type of solutions to systems of conservation laws. J. Differ. Equ. 228(1), 49–86 (2006)
Slemrod, M., Tzavaras, A.E.: A limiting viscosity approach for the Riemann problem in isentropic gas dynamics. Indiana Univ. Math. J. 38(4), 1047–1074 (1989)
Shelkovich, V.M.: Delta-shock waves of a class of hyperbolic systems of conservation laws. In: Abramian, A., Vakulenko, S., Volpert, V. (eds.) Patterns and Waves, pp. 155–168. AkademPrint, St. Petersburg (2003)
Shelkovich, V.M.: The Riemann problem admitting \(\delta \)-, \(\delta '\)-shocks, and vacuum states (the vanishing viscosity approach). J. Differ. Equ. 231(2), 459–500 (2006)
Sheng, W., Zhang, T.: The Riemann problem for transportation equations in gas dynamics. Mem. Amer. Math. Soc., vol. 137(564). Am. Math. Soc., Providence (1999)
Tan, D., Zhang, T., Zheng, Y.: Delta shock waves as limits of vanishing viscosity for hyperbolic system of conservation laws. J. Differ. Equ. 112(1), 1–32 (1994)
Tupciev, V.A.: On the method of introducing viscosity in the study of problems involving the decay of a discontinuity. Dokl. Akad. Nauk SSSR 211, 55–58 (1973). English translation: Sov. Math. Dokl. 14, 978–982 (1973)
Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159(2), 447–484 (1999)
Yang, H., Liu, J.: Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation. Sci. China Math. 58(11), 2329–2346 (2015)
Yang, H., Liu, J.: Concentration and cavitation in the Euler equations for nonisentropic fluids with the flux approximation. Nonlinear Anal. 123–124, 158–177 (2015)
Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252(11), 5951–5993 (2012)
Yang, H., Zhang, Y.: Delta shock waves with Dirac delta function in both components for systems of conservation laws. J. Differ. Equ. 257(12), 4369–4402 (2014)
Yang, H., Zhang, Y.: Flux approximation to the isentropic relativistic Euler equations. Nonlinear Anal. 133, 200–227 (2016)
Zhang, Y., Yang, H.: Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas. J. Math. Anal. Appl. 435(2), 1160–1182 (2016)
Acknowledgements
The authors are very grateful to the anonymous referee for his/her corrections and suggestions, which have improved the original manuscript greatly.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Supported by the National Natural Science Foundation of China 11501488, the Scientific Research Foundation of Xinyang Normal University (No. 0201318) and Nan Hu Young Scholar Supporting Program of XYNU.
Rights and permissions
About this article
Cite this article
Zhang, Y., Zhang, Y. Vanishing Viscosity Limit for Riemann Solutions to a Class of Non-Strictly Hyperbolic Systems. Acta Appl Math 155, 151–175 (2018). https://doi.org/10.1007/s10440-017-0149-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-017-0149-7