Perturbative global solutions of a large class of cross diffusion systems in any dimension

This article focuses on a large family of cross-diffusion systems of the form $\mathrm{\partial }$ t U-$\mathrm{\Delta }$A(U) = 0, in dimension d $\in$ N * , and where U $\in$ R 2. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.