Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion

We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in mean-field limit of interacting stochastic particle systems. To date, their analysis multiple space dimensions has confined the purely convective case equal mobility coefficients. In this article, we introduce normal form for an entropic class such which reveals structure symmetric hyperbolic–parabolic system. Due state-dependence range and kernel singular diffusive matrix, our way rewriting is different from that classically used second-order systems nullspace invariance property. By means change variables, solve Cauchy problem short times positive initial data Hs(Td) s>d/2+1.