Quadratic interaction functional for general systems of conservation laws

TL;DR

This paper introduces a quadratic interaction estimate for systems of conservation laws, extending Glimm's method without structural assumptions on the flux, and provides a new wave tracking and control functional.

Contribution

It develops a novel quadratic interaction estimate for general systems, including a Lagrangian wave representation and a new functional controlling wave speed variation.

Findings

Quadratic interaction estimate with total variation squared bound

Lagrangian wave trajectory representation

Introduction of a new wave speed variation functional

Abstract

For the Glimm scheme approximation $u_\epsilon$ to the solution of the system of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \mathbb R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. $\mathrm{TV}(u_0)$) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux $f$ are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely we obtain the following results: \newline - a new analysis of the interaction estimates of simple waves; \newline - a Lagrangian representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$ which follows the trajectory of each wave $w$ from its creation to its cancellation; \newline - the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; \item a new functional $\mathfrak Q$ controlling the variation in speed of the waves w.r.t. time. \newline This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure with total mass $\leq \mathrm{const} \mathrm{TV}(u_0)^2$.