Rich hyperbolic systems with conservation laws. (Systèmes hyperboliques riches de lois de conservation.) (French) Zbl 0870.35067

Brezis, H. (ed.) et al., Nonlinear partial differential equations and their applications. Collège de France Seminar, volume XI. Lectures presented at the weekly seminar on applied mathematics, Paris, France, 1989-1991. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 299, 248-281 (1994).
This is a detailed study of the “rich” hyperbolic systems, i.e. the diagonal systems of the form \[ \partial_tw_i+ \lambda_i(w_1,w_2,\dots,w_n)\partial_x w_i=0 \] to which any hyperbolic system can be reduced when \(n=2\). The author gives many interesting properties of these systems. The derivative \(\partial_xw_i\) obeys a Riccati equation (like the weak discontinuities of the hyperbolic systems). He gives the construction of rich systems. The linearly degenerated case is also considered.
For the entire collection see [Zbl 0785.00024].


35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations