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Systems describing the long-range interaction between individuals haveattracted a lot of attention in the last years, in particular in relation withliving systems. These systems are quadratic, written under the form oftransport equations with a nonlocal self-generated drift. We establish thelocalisation limit, that is the convergence of nonlocal to local systems, whenthe range of interaction tends to 0. These theoretical results are sustained bynumerical simulations. The major new feature in our analysis is that we do notneed diffusion to gain compactness, at odd with the existing literature. Thecentral compactness result is provided by a full rank assumption on theinteraction kernels. In turn, we prove existence of weak solutions for theresulting system, a cross-diffusion system of quadratic type.

Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion