Small inertia regularization of an anisotropic aggregation model

We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $varepsilon$, which can be interpreted as inertia or response time. We examine rigorously the limit $varepsilon to 0$ 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 $mathcal{O}(varepsilon^{2/3})$. We validate this scale with numerical simulations.