A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux

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. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique.