Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case

TL;DR

This work analyzes an unstable scalar conservation law with gradient-discontinuous flux switching between $f(u)$ and $g(u)$ according to the sign of $u_x$, under the assumption $f>g$ and strict convexity. It develops a BV weak-solution framework with a flux selector $\theta$ and demonstrates nonuniqueness even for smooth data, while providing a complete Riemann-problem classification and constructing globally defined, piecewise monotone solutions for BV initial data. A minimal-interface principle yields a unique global solution by controlling the number of flux-switch interfaces, with interface motion governed by a discontinuous ODE and proven via contraction-type arguments and a specialized uniqueness lemma. The results have implications for traffic-flow modeling (hysteresis-like flux switching) and suggest stochastic extensions where interface creation is randomized, illuminating the mechanism behind stop-and-go dynamics.

Abstract

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible.

Figures (14)

• Figure 1: A solution of (\ref{['1']})-(\ref{['2']}) in the stable case where $f< g$. Here the initial flux is $f(u)$ for $x<x_1$ and $x>x_2$, and $g(u)$ for $x_1<x<x_2$. This produces a sink at $x_1$ and a source at $x_2$.
• Figure 2: A solution of (\ref{['1']})-(\ref{['2']}) in the unstable case where $f> g$. Here the initial flux is $f(u)$ for $x<x_1$ and $x>x_2$, and $g(u)$ for $x_1<x<x_2$. This produces a source at $x_1$ and a sink at $x_2$.
• Figure 3: Left: when $\theta^-\not=\theta^+$, upward jumps can never satisfy the Lax admissibility conditions (\ref{['Lax']}). Right: when $\theta^-\not= \theta^+$, there can also be downward jumps that do not satisfy (\ref{['Lax']}).
• Figure 4: The solution described at (\ref{['bss1']}).
• Figure 5: Left: The tangent lines from the origin to the graph of $f$. Right: the solutions $u_1$ at (\ref{['idex2']}) and $u_2$ at (\ref{['idex3']}).

Theorems & Definitions (17)

• Definition 1.1
• Remark 1.1
• Definition 1.2
• Lemma 2.1
• Remark 2.1
• Lemma 2.2
• proof
• Example 2.1
• Example 2.2
• Definition 3.1