A vanishing dynamic capillarity limit equation with discontinuous flux

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation [Formula: see text] Here, [Formula: see text] and [Formula: see text] are smooth functions while [Formula: see text] and [Formula: see text] are fixed constants. Assuming [Formula: see text] for some [Formula: see text] , strongly as [Formula: see text] , we prove that, under an appropriate relationship between [Formula: see text] and [Formula: see text] depending on the regularity of the flux [Formula: see text] , the sequence of solutions [Formula: see text] strongly converges in [Formula: see text] toward a solution to the conservation law [Formula: see text] The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.