Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0<α<1

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel ϕ(x)=|x|−(1+α). Following our works Shvydkoy and Tadmor (2017) [1], [2] which focused on the range 1≤α<2, and Do et al. (2017) which covered the range 0<α<1, in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in (ρ0,u0)∈H2+α×H3, the solution approaches exponentially fast to a flocking state solution consisting of a wave ρ̄=ρ∞(x−tū) traveling with a constant velocity determined by the conserved average velocity ū. The convergence is accompanied by exponential decay of all higher order derivatives of u.