# Preprint 2013-012

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

## Stefano Bianchini and Stefano Modena

**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.

**References**

[2] F. Ancona, A. Marson,

*Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems*, Comm. Math. Phys.

**302**(2011), 581-630. [MR2774163]

[11] J. Hua, Z. Jiang, T. Yang,

*A New Glimm Functional and Convergence Rate of Glimm Scheme for General Systems of Hyperbolic Conservation Laws*, Arch. Rational Mech. Anal.

**196**(2010), 433-454. [MR2609951]