Wave-front tracking for a quasi-linear scalar conservation law with hysteresis II: the case of Preisach

TL;DR

This paper extends wave-front tracking methods to a quasi-linear scalar conservation law with hysteresis modeled by the Preisach operator, addressing the complexities introduced by internal variables and ensuring solution uniqueness.

Contribution

It introduces a new analysis for the Preisach hysteresis operator in scalar conservation laws, expanding previous work from the Play operator case.

Findings

Developed wave-front tracking for Preisach hysteresis

Established entropy-like condition for uniqueness

Extended analysis to internal variable models

Abstract

We consider the Cauchy problem for the quasi-linear scalar conservation law \[u_t+\mathcal{F}(u)_t+u_x=0,\] where $\mathcal{F}$ is a specific hysteresis operator. Hysteresis models a rate-independent memory relationship between the input $u$ and its output, giving a non-local feature to the equation. In a previous work the authors studied the case when $\mathcal{F}$ is the Play operator. In the present article, we extend the analysis to the case of Preisach operator, which is probably the most versatile mathematical model to describe hysteresis in the applications, especially for the presence of some kind of internal variables. This fact has required a new analysis of the equation. Starting from the Riemann problem, we address the so-called wave-front tracking method for a solution to the Cauchy problem with bounded variation initial data. An entropy-like condition is also exploited for uniqueness.