# Continuous solutions to a balance equation

The talk is concerned with continuous solutions to a scalar, 1D balance law having bounded source term. We will discuss the correspondence between Eulerian and Lagrangian formulation. I will mostly focus on the simple equation $u_{t}+[u^{2}/2]_{x}=g$, $g$ bounded, assuming $u$ continuous but neither Sobolev nor BV. This is strictly related to a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups, after the characterization which had already been given of intrinsic regular graphs. The talk will be mainly based on collaborations with G. Alberti, S. Bianchini, F. Bigolin, F. Serra Cassano.