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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.

MSC:

35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
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