Construction of weak solutions to a model of pressureless viscous flow driven by nonlocal attraction–repulsion

We analyze the pressureless Navier-Stokes system with nonlocal attraction–repulsion forces. Such systems appear in the context of models of collective behaviour. We prove the existence of weak solutions on the whole space $R^3$ in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch–Desjardins and Mellet–Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu [37], [36] to construct a weak solution.