Singular perturbations of first-order hyperbolic systems with stiff source terms. (English) Zbl 0942.35110

This paper deals with initial value problems for nonlinear first-order hyperbolic systems \[ \partial_t U + \sum_{j=1}^{n} A_{j}(U)\partial_{x_j} U = Q(U)/\varepsilon \] where \(U=U(x,t):\mathbb{R}^n\times[0,\infty) \to \mathbb{R}^n\) is the unknown vector function. Here \(0<\varepsilon \ll 1\) is a small parameter, and the nonlinear term \(Q(U)/\varepsilon\) represents a stiff source term. Under a suitable structural stability condition the author constructs formal asymptotic expansions of solutions having an initial layer. This structural condition is equivalent to the well-known subcharacteristic condition for \(2\times 2\) one-dimensional hyperbolic systems. Convergence of asymptotic expansions is proven. The theory developed in this paper can be applied to several problems arising from mathematical physics like e.g. equations of kinetic gas theory, reacting flow models, relativistic magnetohydrodynamic equations etc.


35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
Full Text: DOI Link


[1] Caflisch, R.; Papanicolaou, G.C., The fluid-dynamical limit of a nonlinear model Boltzmann equation, Comm. pure appl. math., 32, 589-616, (1979) · Zbl 0438.76059
[2] Chen, G.-Q.; Levermore, C.D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. pure appl. math., 47, 787-830, (1994) · Zbl 0806.35112
[3] Chen, G.-Q.; Liu, T.-P., Zero relaxation and dissipation limits for hyperbolic conservation laws, Comm. pure appl. math., 46, 755-781, (1993) · Zbl 0797.35113
[4] Eckhaus, W., Matching principles and composite expansions, Lecture notes in math., (1977), Springer-Verlag Berlin, p. 146-177
[5] Geel, R., Singular perturbations of hyperbolic type, (1978), University of AmsterdamMathematisch Centrum · Zbl 0498.35001
[6] Gu, Z.-M.; Nefedov, N.N.; O’Malley, R.E., On singular singularly perturbed initial value problems, SIAM J. appl. math., 49, 1-25, (1989) · Zbl 0676.34038
[7] Van Harten, A.; Van Hassel, R.R., A quasilinear, singular perturbation problem of hyperbolic type, SIAM J. math. anal., 16, 1258-1267, (1985) · Zbl 0612.35007
[8] de Jager, E.M., Singular perturbations of hyperbolic type, Nieuw arch. wisk., 23, 145-172, (1975) · Zbl 0304.35007
[9] Jin, S.; Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. pure appl. math., 48, 235-277, (1995) · Zbl 0826.65078
[10] John, F., Partial differential equations, (1982), Springer-Verlag New York
[11] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. rational mech. anal., 58, 181-205, (1975) · Zbl 0343.35056
[12] Kreiss, H.-O., Problems with different time scales for partial differential equations, Comm. pure appl. math., 33, 399-439, (1980) · Zbl 0439.35043
[13] LeVeque, R.J.; Roe, P.; van Leer, B.; Yee, H.C., Final report, (1989)
[14] Liu, T.-P., Hyperbolic conservation laws with relaxation, Comm. math. phys., 108, 153-175, (1987) · Zbl 0633.35049
[15] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, (1984), Springer-Verlag New York · Zbl 0537.76001
[16] R. Natalini, Recent mathematical results on hyperbolic relaxation problems, in, Analysis of systems of conservation laws, (, H. Freistühler, Ed.), Pitman Research Notes in Mathematics Series, Longman, Harlow, in press.
[17] O’Malley, R.E., Introduction to singular perturbations, (1974), Academic Press New York · Zbl 0287.34062
[18] Platkowski, T.; Illner, R., Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM rev., 30, 213-255, (1988) · Zbl 0668.76087
[19] Rosemann, J.J.; Meyer, R.E., Hyperbolic-hyperbolc systems, J. differential equations, 10, 403-411, (1970)
[20] Schochet, S., Hyperbolic-hyperbolic singular limits, Comm. partial differential equations, 12, 589-632, (1987) · Zbl 0629.35079
[21] Whitham, J., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
[22] Yong, W.-A., Singular perturbations of first-order hyperbolic systems, (1992), Universität Heidelberg
[23] Yong, W.-A., Existence and asymptotic stability of traveling wave solutions of a model system for reacting flow, Nonlinear anal., 26, 1791-1809, (1996) · Zbl 0873.76092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.