Quadratic interaction functional for systems of conservation laws: a case study

We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws \[ \left\{ \begin{array}{c} u_t + \tilde f(u,v)_x = 0, \\ v_t - v_x = 0. \end{array} \right. \] This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme \cite{anc_mar_11_CMP}. Our aim is to extend the analysis, done for scalar conservation laws \cite{bia_mod_13}, 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. %cancellations it variation is controlled by the total variation growths at most of the total variation of the solution multiplied by the amount of cancellation. 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 $(u(t,x),v(t,x))$ and the reconstruction of the past history of each wavefront involved in an interaction.