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Stability of a resonant system of conservation laws modeling polymer flow with gravitation. (English) Zbl 0977.35083
The authors prove existence, uniqueness, and \(L^1\)-stability of entropy solutions to the Cauchy problem for the following resonant \(2\times 2\) system of conservation laws \[ s_t+f(s,c)_x=0, \quad (sc)_t+(cf(s,c))_x=0. \] Such system arises as a model for two phase polymer flow in porous media. The methods are based on front tracking approximations for the auxiliary scalar equation and the Kruzhkov’s entropy condition for scalar conservation laws.

MSC:
35L65 Hyperbolic conservation laws
76S05 Flows in porous media; filtration; seepage
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[1] Baiti, P; Jenssen, H.K, Well-posedness for a class of conservation laws with l∞ data, J. differential equations, 140, 161-185, (1997) · Zbl 0892.35097
[2] Bressan, A; Colombo, R, The semigroup generated by 2×2 systems of conservation laws, Arch. rational mech., 133, 1-75, (1995) · Zbl 0849.35068
[3] Dahl, O; Johansen, T; Tveito, A; Winther, R, Multicomponent chromatography in a two phase enviroment, SIAM J. appl. math., 52, 65-104, (1992) · Zbl 0739.35064
[4] Gimse, T; Risebro, N.H, Riemann problems with discontinuous flux functions, Third international conference on hyperbolic problems, theory numerical methods and applications, (1991), Studentlitteratur/Chartwell-Bratt, p. 488-452 · Zbl 0789.35102
[5] Gimse, T; Risebro, N.H, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. math. anal, 23, 635-648, (1992) · Zbl 0776.35034
[6] Holden, H; Holden, L; Høegh-Krohn, R, A numerical method for first order nonlinear scalar conservation laws, Comput. math. appl., 15, 595-602, (1988) · Zbl 0658.65085
[7] E. Isaacson, Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enahanced oil recovery, preprint, Rockefeller University, New York, 1981.
[8] Isaacson, E; Temple, B, Nonlinear resonance in systems of conservation laws, SIAM J. appl. math., 52, 1260-1278, (1992) · Zbl 0794.35100
[9] Isaacson, E; Temple, B, Convergence of the 2× Godunov method for a general resonant nonlinear balance law, SIAM J. appl. math., 55, 625-640, (1995) · Zbl 0838.35075
[10] H. K. Jenssen, private communication.
[11] K. Karlsen, and, N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. Available at, http://www.math.ntnu.no/conservation/2000/035.ps. · Zbl 1032.76048
[12] R. A. Klausen, and, N. H. Risebro, Well posedness of a 2×2 system of resonant conservation laws, in, Proc. of the 8th International Conference on Hyperbolic Problems, Zürich, 1998. · Zbl 0931.35094
[13] Klausen, R.A; Risebro, N.H, Stability of conservation laws with discontinuous coefficients, J. differential equations, 157, 41-60, (1999) · Zbl 0935.35097
[14] Klingenberg, C; Risebro, N.H, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymtotic behavior, Comm. partial differential equations, 20, 1959-1990, (1995) · Zbl 0836.35090
[15] Kružkov, S.N, First order quasilinear equation in several independent variables, Mat. sb., 10, 217-243, (1970) · Zbl 0215.16203
[16] Kuznetsov, N.N, Accuracy of some approximate methods for computing the weak solutions of a first order quasilinear equation, USSR comput. math. math. phys., 16, 105-119, (1976) · Zbl 0381.35015
[17] Longwai, L; Temple, B; Jinghua, W, Suppression of oscillations in Godunov’s method for a resonant non-strictly hyperbolic system, SIAM J. numer. anal., 32, 841-864, (1995) · Zbl 0845.65052
[18] L. Longwai, B. Temple, and, J. Wang, Uniqueness and continuous dependence for a resonant non-strictly hyperbolic system, unpublished note.
[19] Oleinik, O.A, Discontinuous solutions of nonlinear differential equations, Trans. amer. math. soc., 26, 95-172, (1963) · Zbl 0131.31803
[20] Smoller, J, Shock waves and reaction-diffusion equations, (1995), Springer-Verlag New York
[21] Temple, B, Global solution of the caucy problem for a 2×2 non-strictly hyperbolic system of conservation laws, Adv. appl. math., 3, 335-375, (1982)
[22] Tveito, A; Winther, R, Existence, uniqueness and continuous dependence for a system of conservation laws modeling polymer flooding, SIAM J. math. anal., 22, 905-933, (1991) · Zbl 0741.65071
[23] Wagner, D, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. differential equations, 68, 118-136, (1987) · Zbl 0647.76049
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