Well-posedness and Godunov approximation for nonlinear conservation laws with hysteresis

This article studies the Cauchy problem for the scalar conservation law (Formula presented.) where w(x,t)=[F(u)(x,t)] is the output of a specific hysteresis operator, namely the Play hysteresis operator, and f is a C2 convex flux function. The hysteresis operator models a rate-independent memory effect, introducing a specific non-local feature into the partial differential equation. We consider 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 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.