Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview

Abstract

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws

\( \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. \)

where u : [0, ∞) × ℝ → ℝⁿ, f : ℝⁿ → ℝⁿ is strictly hyperbolic, and Tot.Var.(u₀) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form

\( \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, \)

where α_j and \( {\alpha}_j^{\prime } \) 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:

• all the main ideas of the construction are presented;

• all the technicalities of the proof in the general setting [8] are avoided.