Eulerian, Lagrangian and broad continuous solutions to a balance law with non-convex flux I

We discuss different notions of continuous solutions to the balance law ∂_tu + ∂_x (f(u))=g, g bounded, f in C^2 extending previous works relative to the flux f(u)=u^2. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also reduce the ODE on any characteristics under the sharp assumption that the set of inflection points of the flux f is negligible. The correspondence of the source terms in the two settings is a matter of the companion work [2], where we include counterexamples when the negligibility on inflection points fails.