Uniqueness for discontinuous O.D.E. and conservation laws

Consider a scalar O.D.E. of the form ẋ = f(t, x), where f is possibly discontinuous w.r.t. both variables t, x. Under suitable assumptions, we prove that the corresponding Cauchy problem admits a unique solution, which depends Hölder continuously on the initial data. Our result applies in particular to the case where f can be written in the form f(t, x) . = g(u(t, x)), for some function g and some solution u of a scalar conservation law, say ut +F (u)x = 0. In turn, this yields the uniqueness and continuous dependence of solutions to a class of 2× 2 strictly hyperbolic systems, with initial data in L.