Deterministic particle approximation of aggregation diffusion equations with nonlinear mobility

We consider a class of aggregation-diffusion equations on unbounded one-dimensional domains with Lipschitz nonincreasing mobility function. We show strong L1-convergence of a suitable deterministic particle approximation to weak solutions of a class aggregation-diffusion PDEs (coinciding with the classical ones in the no vacuum regions) for any bounded initial data of finite energy. In order to prove well-posedness and convergence of the scheme with no BV or no vacuum assumptions and overcome the issues posed in this setting by the presence of a mobility function, we improve and strengthen the techniques introduced in [S. Daneri, E. Radici and E. Runa, Deterministic particle approximation of aggregation-diffusion equations on unbounded domains, J. Differential Equations 312 (2020) 474-517].