Small Inertia Regularization Of An Anisotropic Aggregation Model

We consider an anisotropic first-order ODE aggregation model in two dimensions and its approximation by a second-order relaxation system. The relaxation model contains a small parameter epsilon, which can be interpreted as inertia or (non-dimensionalized) response time. We examine rigorously the limit epsilon -> 0 of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities, solutions to the second-order model perform fast transitions within a time layer of size O(epsilon(2/3)). We validate this scale with numerical simulations.