Wave-front tracking for a quasi-linear scalar conservation law with hysteresis

Abstract

In this article we deal with the Cauchy problem for the quasi-linear scalar conservation law \[u_t+ {\cal F}(u)_t+u_x=0,\] where ${\cal F}$ is a specific hysteresis operator, namely the Play operator. Hysteresis models a rate-independent memory relationship between the input $u$ and its output. Its presence in the partial differential equation gives a particular non-local feature to the latter allowing us to capture phenomena that may arise in some application fields. Riemann problems and the interactions between shock lines are studied and via the so-called Wave-Front Tracking method a solution to the Cauchy problem with bounded variation initial data is constructed. The solution found satisfies an entropy-like condition, making it the unique solution in the class of entropy admissible ones.