Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
Authors:
L. Hsiao and P. de Mottoni
Journal:
Trans. Amer. Math. Soc. 322 (1990), 121158
MSC:
Primary 35L65; Secondary 35M10
MathSciNet review:
938919
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Abstract: We study the system of conservation laws given by with any Riemann initial data . The system is elliptic in the domain where and strictly hyperbolic when . We combine and generalize Lax criterion and OleinikLiu criterion to introduce the generalized entropy condition (G.E.C.) by which we can show that the Riemann problem always has a weak solution (any discontinuity satisfies the G.E.C.) for any initial data, however not necessarily unique. We introduce the minimum principle then in the definition of an admissible weak solution for the Riemann problem and the existence and uniqueness of the solution for any Riemann data.
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 [Ha]
 H. Hattori, The Riemann problem for a Van der Waals fluid with entropy rate admissibility criterionIsothermal case, Arch. Rational Mech. Anal. (to appear). MR 837688 (87f:35221b)
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 H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure. Appl. Math. 40 (1987), 229264. MR 872386 (88d:35125)
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 , Qualitative behavior of solutions for Riemann problems of conservation laws of mixed type, Proc. of the Second International Conference on Nonlinear Hyperbolic Problems, Aachen, FRG, March 1418, 1988, Notes on Numerical Fluid Mechanics, Vol. 24, 1989, pp. 246256. MR 991370 (90h:35154)
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 L. Hasiao and P. de Mottoni, Quasilinear hyperbolic system of conservation laws with parabolic degeneracy, Rocky Mountain J. Math. (to appear).
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 R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73 (1980), 125158. MR 556559 (80k:73009)
 [K]
 B. L. Keyfitz, The Riemann problem for nonmonotone stressstrain functions: A "Hysteresis" approach (to appear). MR 837687 (87f:35221a)
 [L]
 T. P. Liu, The Riemann problem for general conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89112. MR 0367472 (51:3714)
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 J. D. Murray and J. E. R. Cohen, On nonlinear convective dispersal effects in an interacting population model, SIAM. J. Appl. Math. 43 (1983), 6678. MR 687790 (84i:92079)
 [Se]
 M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Differential Equations 46 (1982), 426443. MR 681232 (84a:35164)
 [Sl]
 M. Slemrod, Admissibility criteria for propagating phase boundaries in a Van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), 301315. MR 683192 (84a:76030)
 [SS]
 D. G. Schaeffer and M. Shearer, Riemann problem for nonstrictly hyperbolic systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), 267306. MR 906816 (88m:35101)
 [T]
 B. Temple, Global solutions of the Cauchy problem for a class of nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982), 335375. MR 673246 (84f:35091)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009389193
PII:
S 00029947(1990)09389193
Article copyright:
© Copyright 1990
American Mathematical Society
