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

We address the Cauchy problem for a scalar nonlinear conservation law where the time derivative accounts for both a primary variable and the output of a specific hysteresis operator, namely the Play hysteresis operator, applied to that variable. 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 the Riemann problem. Furthermore, a Godunov-type finite volume numerical scheme is developed to compute approximate solutions. The convergence of the scheme for initial data of bounded variation 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.