  This article focuses on a large family of cross-diffusion systems of the form
$\partial$ t U-$\Delta$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.
