Neutral functional differential equations and systems of conservation laws

Abstract

A natural way of introducing neutral functional differential equations is integration along the characteristics of the Riemann invariants of the hyperbolic partial differential equations. Up to now the problem has been discussed by A.D. Myshkis, K.L. Cooke and their followers in the linear or quasilinear case. The conservation laws are nonlinear hyperbolic partial differential equations whose study goes back to the fifties of the previous century, being due to the pioneering papers of O.A. Oleinik and P.D. Lax. These equations describe important phenomena in Physics and Engineering, being also subject to control issues. The present paper is an attempt to extend the method of integrating along the characteristics to the systems of conservation laws. The main purpose of this attempt is to emphasize the occurring nonlinear neutral functional differential equations and to construct the augmented validation (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level of the classical solutions.