Abstract: We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws $$ \left\{ \begin{aligned} u_t +\tilde f(u,v)_x &=0\\ v_t − v_x &= 0. \end{aligned} \right. $$ This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme [2].

Our aim is to extend the analysis, done for scalar conservation laws [7], in the presence of transversal interactions among wavefronts of different families. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its total variation in time is bounded.

The study of this particular system is a key step in the proof of the quadratic interaction estimate for general systems: it requires a deep analysis of the wave structure of the solution $\bigl(u(t, x), v(t, x)\bigr)$ and the reconstruction of the past history of each wavefront involved in an interaction.