When do cross-diffusion systems have an entropy structure?

In this note, necessary and sufficient conditions for the existence of an entropy structure certain classes cross-diffusion systems with diffusion matrix A ( u ) are given, based on results from factorization. The is important in analysis such equations since typically neither symmetric nor positive definite. particular, normal ellipticity all symmetry Onsager implies its definiteness hence structure. If constant or up to nonlinear perturbations, equivalent . applied various examples physics biology. Finally, n -species population model Shigesada, Kawasaki, Teramoto proved.