Basic aspects of hyperbolic relaxation systems.(English)Zbl 1017.35068

Freistühler, Heinrich (ed.) et al., Advances in the theory of shock waves. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 47, 259-305 (2001).
This paper is devoted to the system of first-order partial differential equations with a small parameter $$\varepsilon>0$$: $U_t+ \sum^d_{j=1} F_j(U)_{x_j}=Q(u)/ \varepsilon\tag{1}$ or more generally $U_t+ \sum^d_{j=1} A_j(U)U_{x_j}=Q(u)/ \varepsilon.\tag{2}$ Starting with Neumann’s stability analysis, the author identifies several basic structural conditions aiming at the existence of a well-behaved limit as $$\varepsilon\to 0$$. Moreover the author studies two typical and two physical relaxation models and presents the result of the zero relaxation limit for initial value problems of nonlinear systems with smooth initial data.
For the entire collection see [Zbl 0966.00009].

MSC:

 35L60 First-order nonlinear hyperbolic equations 35B25 Singular perturbations in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems