Existence theory for hyperbolic systems of conservation laws with general flux-functions

For the Cauchy problem associated with a nonlinear, strictly hyperbolic system of conservation laws in one space dimension, we establish a general existence theory in the class of functions with sufficiently small total variation (say less than some constant c). To begin with, we assume that the flux-function f(u) is piecewise genuinely nonlinear, in the sense that it exhibits finitely many (at most p, say) points of lack of genuine nonlinearity along each wave curve. Importantly, our analysis applies to arbitrary large p, in the sense that the constant c restricting the total variation is independent of p. Second, by an approximation argument, we prove that the existence theory above extends to general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f^ε(u). The main contribution in this paper is the derivation of uniform estimates for the wave curves and wave interactions (which are entirely independent of the properties of the flux-function) together with a new wave interaction potential which is decreasing in time and is a fully local functional depending upon the angle made by any two propagating discontinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p = p(v) satisfying solely the hyperbolicity condition p′(v) < 0 but no convexity assumption.