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

This article studies the Cauchy problem for the scalar conservation law ∂ ₜ u + ∂ ₜ w + ∂ ₓ f(u) = 0, ∂ t u + ∂ t w + ∂ x f ( u ) = 0 , where w(x,t) = [F(u)(x,t)] w ( x , t ) = [ F ( u ) ( x , t ) ] is the output of a specific hysteresis operator, namely the Play hysteresis operator, and f f is a C² 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 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 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.