Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws u_t+f(u)_x=0, u(t=0)=u_0(x), where u : [0, ∞) × R → R^n, f : R^n → R^n is strictly hyperbolic, and Tot.Var.(u_0) << 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form \sum_{tj interaction time} |σ(α_j)−σ(α_j′)||α_j||α_j′||α_j|+|α_j′|≤C(f)Tot.Var.(u_0)^2, where α_j and α_j′ are the wavefronts interacting at the interaction time t_j, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: a) all the main ideas of the construction are presented; b) all the technicalities of the proof in the general setting [8] are avoided.