A Uniqueness Condition for Nonlinear Convection-diffusion Equations with Discontinuous Coefficients

Abstract. The paper focuses on the uniqueness issue for scalar convection-diffusion equations where both the convective flux and diffusion functions have a spatial discontinuity. An interface entropy condition is proposed at such a spatial discontinuity. It implies the Kružkov-type entropy condition presented by Karlsen et al. [Trans. Royal Norwegian Society Sci. Letters 3, 49 pp, 2003]. They proved uniqueness when the convective flux function satisfies an additional ‘crossing condition’. The crossing condition becomes redundant with the entropy condition proposed here. Thereby, more general flux functions are allowed. Another advantage of the entropy condition is its simple geometrical interpretation, which facilitates the construction of stationary solutions.