Well-posedness and numerical approximation of nonlinear conservation laws with hysteresis

TL;DR

This paper establishes the well-posedness of a nonlinear conservation law with hysteresis effects, introduces a suitable entropy solution concept, and develops a convergent numerical scheme for approximating solutions.

Contribution

It provides the first rigorous analysis of a conservation law with hysteresis, defining entropy solutions and proving convergence and stability of a numerical scheme.

Findings

Existence of entropy weak solutions for the problem.

Convergence of a Godunov-type scheme for BV initial data.

Uniqueness and stability of solutions are established.

Abstract

This article studies the Cauchy problem for the scalar conservation law \[ \partial_t u + \partial_t w + \partial_x f(u) = 0, \] where $w(x,t) = [\mathcal{F}(u)(x,t)]$ is the output of a specific hysteresis operator, namely the Play hysteresis operator, and $f$ is a $\mathbf{C}^2$ convex flux function. The hysteresis operator models a rate-independent memory effect, introducing a specific non-local feature into the partial differential equation. We define a suitable notion of entropy weak solution and analyse in detail the Riemann problem. Furthermore, a Godunov-type finite volume numerical scheme is developed to compute approximate solutions. The convergence of the scheme for $\mathrm{BV}$ initial data provides the existence of an entropy weak solution. Finally, a stability estimate is established, implying the uniqueness and overall well-posedness of the entropy weak solution.