  The uniqueness of bounded weak solutions to strongly coupled parabolic
equations in a bounded domain with no-flux boundary conditions is shown. The
equations include cross-diffusion and drift terms and are coupled
selfconsistently to the Poisson equation. The model class contains special
cases of the Maxwell-Stefan equations for gas mixtures, generalized
Shigesada-Kawasaki-Teramoto equations for population dynamics, and
volume-filling models for ion transport. The uniqueness proof is based on a
combination of the $H^{-1}$ technique and the entropy method of Gajewski.
