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

TL;DR

This paper addresses the well-posedness and numerical approximation of a scalar conservation law with a rate-independent memory effect modeled by the Play hysteresis operator: $\partial_t u + \partial_t w + \partial_x f(u) = 0$ with $w = [\mathcal{F}(u,w_0)]$ and a convex flux $f$. The authors develop an entropy weak solution framework, analyze the Riemann problem to characterize admissible wave patterns, and introduce a Godunov-type finite-volume scheme that converges to the entropy solution for BV initial data. Key contributions include a constructive existence proof via a convergent numerical scheme, discrete entropy and weak hysteresis conditions, and a stability estimate yielding uniqueness. The results provide a rigorous foundation and practical numerical method for nonlinear conservation laws with hysteresis, capturing the memory effects intrinsic to Play-type hysteresis and enabling reliable simulations of systems with rate-independent memory.

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.

Figures (18)

• Figure 1: The Play hysteresis operator
• Figure 2: An explicit example of the operator $\mathcal{F}$ applied to $u$ with a jump discontinuity. Here $a=1$, $u=u_-=0$ for $t<t^*$, $u=u_+=2$ for $t>t^*$ and $w_0=0$; consequently $w=w_0$ for $t<t^*$ and $w=w^*=1$ for $t>t^*$. In red we highlight the path followed by the couple $(u_\varepsilon,w_\varepsilon)$ which, after the limiting procedure, collapses to $(u_-,w_0)$ for $t<t^*$ and $(u_+,w^*)$ for $t>t^*.$
• Figure 3: An example of $\bar{f}_{w_r}$ (left) and $\hat{f}_{w_l}$ (right) compared with $f$ (gray); the additive constants $\tfrac{1}{2} f(w_r-a)$ in \ref{['eq: fbar']} and $\tfrac{1}{2} f(w_r+a)$ in \ref{['eq: fhat']} are needed for the continuity of $\bar{f}_{w_r}$ and $\hat{f}_{w_l}$ respectively.
• Figure 4: On the left, the couple $(u,w)$ is shown in the hysteresis plane: the arrow represents the path followed by $t\mapsto (u(x,t),w(x,t))$ for $x>0$; the dashed line instead represents the stationary shock. On the right, the solution $(u,w)$ in the $(x,t)$ plane: it consists of two rarefaction waves and a stationary shock for $w$ along $x = 0$. In this example, the flux function is $f(u) = \frac{1}{2} u^2$, the parameter $a = 1$, and the initial data $(u_l,w_l)=(1,0.5)$ and $(u_r,w_r)=(3,3).$
• Figure 5: An example of the case when $f'(u_r) \leq 0$. In this example, the flux function is $f(u) = \frac{1}{2} u^2$, the parameter $a = 1$, and the initial data $(u_l,w_l)=(-3,-3)$ and $(u_r,w_r)=(-1.5,-1)$.

Theorems & Definitions (35)

• Proposition 1.1
• Definition 1.2
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 3.1
• proof
• Proposition 3.2