Entropy solutions in $B{V}^s$ for a class of triangular systems involving a transport equation

In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in $L^{\mathrm{\infty }}$ under optimal fractional $BV$ regularity of the initial data. Our results show that the critical fractional regularity is $s=1\mathrm{/}3$. We also construct an initial data that is not in $B{V}^{1\mathrm{/}3}$ but for which a blow-up in $L^{\mathrm{\infty }}$ occurs, proving the optimality of our results.