Abstract
Substantial progresses have been made in recent years on shock wave theory. The present article surveys the exact mathematical theory on the behavior of nonlinear hyperbolic waves and raises open problems.
This paper was written while the author was visiting the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455
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References
Chern, I.-L., and Liu, T.-P., Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), 503–517.
Di Perna, R., Uniqueness of solutions to hyperbolic conservation laws, Indiana U. Math. J. 28 (1979), 244–257.
Glimm, J., Solutions in the large for nonlinear systems of equations, Comm. Pure Appl. Math., 18 (1965), 95–105.
Glimm, J. and Lax, P.D., Decay of solutions of nonlinear hyperbolic conservation laws, Amer. Math. Soc. Memoir, No. 101 (1970).
Goodman, J. and Xin, Z., (preprint).
Hopf, D. and Liu, T.-P., The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data, Indiana U. J. (1989).
Hsu, S.-B. and Liu, T.-P., Nonlinear singular Sturm-Liouville problem and application to transonic flow through a nozzle, Comm. Pure Appl. Math. (1989).
Lax, P.D., Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957), 537–566.
Lin, L., On the vacuum state for the equation of isentropic gas dynamics, J. Math. Anal. Appl. 120 (1987), 406–425.
Liu, T.-P., Deterministic version of Glimm scheme, Comm. Math. Phys., 57 (1977), 135–148.
Liu, T.-P., Linear and nonlinear large time behavior of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767–798.
Liu, T.-P., Admissible solutions of hyperbolic conservation laws, Amer. Math. Soc. Memoir, No. 240 (1981).
Liu. T.-P., Nonlinear stability of shock waves for viscous conservation laws, Memoirs, Amer. Math. Soc. No. 328 (1975).
Liu, T.-P., Hyperbolic conservation laws with relaxation, Math. Phys. 108 (1987), 153–175.
Liu, T.-P., and Xin, Z., Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, 118 (1988), 415–466.
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© 1991 Springer-Verlag New York, Inc.
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Liu, TP. (1991). Geometric Theory of Shock Waves. In: Glimm, J., Majda, A.J. (eds) Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9121-0_16
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DOI: https://doi.org/10.1007/978-1-4613-9121-0_16
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