  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.
