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 vari-ation (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 nonlin-earity along each wave curve. Importantly, our analysis applies arbitrary large p, in the sense that the constant c restricting the total variation is in-dependent of p. Second, by an approximation argument, we prove that the existence theory above extends to more general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f(u). The main contribution in this paper is the derivation of uniform esti-mates for the wave curves and wave interactions (which are entirely inde-pendent 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 discon-tinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p = p(v) satisfying solely the hyperbol-icity condition p′(v) < 0 but no convexity assumption. 1