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 [Formula: see text] or [Formula: see text], when the gradient [Formula: see text] of the solution is positive or negative, respectively. In the stable case where [Formula: see text] for all [Formula: see text], it was proved in [D. Amadori, A. Bressan and W. Shen, Conservation laws with discontinuous gradient-dependent flux: The stable case, Math. Models Methods Appl. Sci. 35(6) (2025) 1421–1469, https://doi.org/10.1142/S0218202525500216 ] that the limits of vanishing viscosity approximations form a contractive semigroup with respect to the [Formula: see text] distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In this 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.