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A convergent series expansion for hyperbolic systems of conservation laws. (English) Zbl 0599.35103
We consider the discontinuous piecewise analytic initial value problem for a wide class of conservation laws that includes the full three- dimensional Euler equations. The initial interaction at an arbitrary curved surface is resolved in time by a convergent series. Among other features the solution exhibits shock, contact, and expansion waves as well as sound waves propagating on characteristic surfaces. The expansion waves correspond to the one-dimensional rarefactions but have a more complicated structure. The sound waves are generated in place of zero strength shocks, and they are caused by mismatches in derivatives.

35L65 Hyperbolic conservation laws
35A10 Cauchy-Kovalevskaya theorems
35A20 Analyticity in context of PDEs
Full Text: DOI
[1] Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. · Zbl 0268.35062
[2] R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. · Zbl 0365.76001
[3] L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin, 1963. · Zbl 0108.09301
[4] Marvin Shinbrot and Robert R. Welland, The Cauchy-Kowalewskaya theorem, J. Math. Anal. Appl. 55 (1976), no. 3, 757 – 772. · Zbl 0346.35002 · doi:10.1016/0022-247X(76)90079-2 · doi.org
[5] A. Majda, The stability of multi-dimensional shock fronts and The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. Nos. 275 and 281, 1983. · Zbl 0517.76068
[6] Принципы современной математической физики, ”Мир”, Мосцощ, 1982 (Руссиан). Транслатед фром тхе Енглиш бы В. Е. Кондрашов, В. Ф. Курякин анд В. Г. Подвал\(^{\приме}\)ный; Транслатион едитед анд щитх а префаце бы И. Д. Софронов.
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