Weak-strong uniqueness for a specific class of cross-diffusion models with volume filling

The weak-strong uniqueness for solutions to a special class of parabolic cross-diffusion systems with volume filling in a bounded domain with no-flux boundary conditions is proved. The diffusion matrix is neither symmetric nor positive definite, but the system possesses a formal gradient-flow or entropy structure. It is shown that any weak solution coincides with a “strong” solution with the same initial data, as long as the “strong” solution exists. The proof is mainly based on the use of the relative entropy modified by small parameters ε and δ, combined with some analytical techniques.