Regular solutions to the fractional Euler alignment system in the Besov spaces framework

We here construct (large) local and small global-in-time regular unique solutions to the fractional Euler alignment system in the whole space ${\mathbb R}^d$, in the case where the deviation of the initial density from a constant is sufficiently small. Our analysis strongly relies on the use of Besov spaces of the type $L^1(0,T;{\dot{B}}_{p,1}^s)$, which allow to get time independent estimates for the density even though it satisfies a transport equation with no damping. Our choice of a functional setting is not optimal but aims at providing a transparent and accessible argumentation.