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 Rd, in the case where the deviation ofthe initial density from a constant is sufficiently small. Our analysis strongly relies on the use of Besov spaces of the type L1(0, T ; ̇Bsp,1),which allow to get time independent estimates for thedensity even though it satisfies a transport equation with nodamping. Our choice of a functional setting is not optimal but aims at providing a transparent and accessible argumentation.