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

Yong, Wen-An

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].

Reviewer: Messoud Efendiev (Berlin)

MSC:

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

Keywords:

Neumann’s stability analysis; structural conditions