Calculus of Variations and Geometric Measure Theory

G. Alberti - S. Bianchini - L. Caravenna

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

created by caravenna on 15 Dec 2015
modified on 12 Oct 2016


Published Paper

Inserted: 15 dec 2015
Last Updated: 12 oct 2016

Journal: Journal of Differential Equations
Volume: 261
Number: 8
Pages: 4298–4337
Year: 2016
Doi: 10.1016/j.jde.2016.06.026


We discuss different notions of continuous solutions to the balance law \[u_t+(f(u)_x=g\] with $g$ 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 fluxes which are merely smooth. We also establish the ODE reduction 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 matter of a companion work, where we also provide counterexamples when the negligibility on inflection points fails.