Quadratic interaction estimate for hyperbolic conservation laws: an overview

S. Modena

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, Tome 59 (2016), pp. 148-172

Résumé

In the joint work with S. Bianchini [8] (see also [6,7]), we proved a quadratic interaction estimate for the system of conservation laws \begin{equation*} \begin{cases} u_t+f(u)_x=0,\\ u(t=0)=u_0(x), \end{cases} \end{equation*} where $u\colon[0,\infty)\times\mathbb R\to\mathbb R^n$, $f\colon\mathbb R^n\to\mathbb R^n$ is strictly hyperbolic, and $\operatorname{Tot.Var.}(u_0)\ll1$ For a wavefront solution in which only two wavefronts at a time interact, such estimate can be written in the form \begin{equation*} \sum_{\text{время взаимодействия }t_j}\frac{|\sigma(\alpha_j)-\sigma(\alpha'_j)||\alpha_j||\alpha'_j|}{|\alpha_j|+|\alpha'_j|}\leq C(f)\operatorname{Tot.Var.}(u_0)^2, \end{equation*} where $\alpha_j$ and $\alpha'_j$ are the wavefronts interacting at the interaction time $t_j,$ $\sigma(\cdot)$ is the speed, $|\cdot|$ 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 of such 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.