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 kernelφ(x) = |x|−(1+α). Following ourworks 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 awave ρ̄ = ρ∞(x−tū) travelingwith a constant velocity determined by the conserved average velocity ū. The convergence is accompanied by exponential decay of all higher order derivatives of u. © 2017 Elsevier B.V. All rights reserved.