Nonlocal-to-Local Limits and Diffusion Approximation in Aggregation Models With Finite-Range Interactions

We investigate the nonlocal-to-local limit of aggregation–diffusion equations with finite-range interactions. The drift is generated by an odd, short-range vector kernel acting on a nonlinear density response. On the flat torus, the rescaled operator is written as a difference quotient and shown to be uniformly bounded from H1 to L2. Under an explicit uniform-density bound and a coercivity condition, compactness of weak solutions follows, and the nonlocal drift converges weakly in L2 to a local gradient field. A strong–weak argument then identifies the nonlinear flux and yields a nonlinear diffusion equation in the small-radius limit. The periodic formulation isolates the bulk limit from boundary layers produced by truncated kernels on bounded domains. For smooth solutions and sufficiently symmetric kernels, a formal R2 correction is derived in terms of the fourth kernel moment. Linear stability analysis shows how the interaction range modifies mode selection before the local regime is reached. A formal individual-based interpretation connects finite-range particle interactions to the continuum equation without assuming a propagation-of-chaos result. Reproducible one-dimensional computations quantify radius convergence and mesh error, while two-dimensional heat maps and three-dimensional space–time surfaces illustrate the evolution and localization of finite-range effects.MSC2020 Classification: 35K55, 35Q92, 45K05, 65M12