Global weak solutions for a two-species Riesz cross-attraction system with quadratic diffusion

We establish global existence for a two-species cross-attraction system on ${\mathbb{R}}^3$, with quadratic diffusion and the common interaction operator $(-\mathrm{\Delta }{)}^{-s}$, where $1\mathrm{/}2<s<1$. Nonnegative initial data in $L^1\cap L^{\mathrm{\infty }}$ with finite second moments require no smallness condition on the masses. A joint energy and second-moment estimate and a coupled $L^p$ iteration give uniform a priori estimates for the regularized solutions. Compactness then yields mass-conserving global weak solutions, bounded on every finite time interval and strongly continuous in $L^1\cap L^2$. The energy inequality holds at every pair of time endpoints. Uniqueness and bounds uniform for all positive times are not asserted.