New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function $f$ is strictly convex and show that, for every $x \in \mathbb{R}$, the total variation of the composite function $f \circ u(\cdot, x)$ is controlled by the total variation of the initial datum. Next, we assume that $f$ is monotone and, under no convexity assumption, we show that, for every $x$, the total variation of the left and right trace $u(\cdot, x^\pm)$ is controlled by the total variation of the initial datum. We also exhibit a counter-example showing that in the first result the total variation bound does not extend to the function $u$, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counter-example due to Bressan and Yu. We also establish stability and propagation of $BV$ regularity, and this is again interesting in view of recent counter-examples.