Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
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- by L. Hsiao and P. de Mottoni PDF
- Trans. Amer. Math. Soc. 322 (1990), 121-158 Request permission
Abstract:
We study the system of conservation laws given by \[ \left \{ {_{{\upsilon _t} + {{[\upsilon (a + u)]}_x} = 0\quad (a > 1{\text {is}}{\text {a}}{\text {constant}}),}^{{u_t} + {{[u(1 - \upsilon )]}_x} = 0,}} \right .\] with any Riemann initial data $({u_ \mp },{\upsilon _ \mp })$. The system is elliptic in the domain where ${(\upsilon - u + a - 1)^2} + 4(a - 1)u < 0$ and strictly hyperbolic when ${(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0$. We combine and generalize Lax criterion and Oleinik-Liu 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.References
-
J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math. 18 (1961), 191-204.
- Barbara Lee Keyfitz, The Riemann problem for nonmonotone stress-strain functions: a “hysteresis” approach, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 379–395. MR 837687
- Helge Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math. 40 (1987), no. 2, 229–264. MR 872386, DOI 10.1002/cpa.3160400206
- L. Hsiao, Admissible weak solution for nonlinear system of conservation laws in mixed type, J. Partial Differential Equations 2 (1989), no. 1, 40–58. MR 1026085
- L. Hsiao, Qualitative behavior of solutions for Riemann problems of conservation laws of mixed type, Nonlinear hyperbolic equations—theory, computation methods, and applications (Aachen, 1988) Notes Numer. Fluid Mech., vol. 24, Friedr. Vieweg, Braunschweig, 1989, pp. 246–256. MR 991370, DOI 10.1080/07468342.1989.11973242 L. Hasiao and P. de Mottoni, Quasilinear hyperbolic system of conservation laws with parabolic degeneracy, Rocky Mountain J. Math. (to appear).
- Richard D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73 (1980), no. 2, 125–158. MR 556559, DOI 10.1007/BF00258234
- Barbara Lee Keyfitz, The Riemann problem for nonmonotone stress-strain functions: a “hysteresis” approach, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 379–395. MR 837687
- Tai Ping Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89–112. MR 367472, DOI 10.1090/S0002-9947-1974-0367472-1
- J. D. Murray and J. E. R. Cohen, On nonlinear convective dispersal effects in an interacting population model, SIAM J. Appl. Math. 43 (1983), no. 1, 66–78. MR 687790, DOI 10.1137/0143006
- Michael Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Differential Equations 46 (1982), no. 3, 426–443. MR 681232, DOI 10.1016/0022-0396(82)90103-6
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI 10.1007/BF00250857
- David G. Schaeffer and Michael Shearer, Riemann problems for nonstrictly hyperbolic $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), no. 1, 267–306. MR 906816, DOI 10.1090/S0002-9947-1987-0906816-5
- Blake Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math. 3 (1982), no. 3, 335–375. MR 673246, DOI 10.1016/S0196-8858(82)80010-9
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 322 (1990), 121-158
- MSC: Primary 35L65; Secondary 35M10
- DOI: https://doi.org/10.1090/S0002-9947-1990-0938919-3
- MathSciNet review: 938919