Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0<\alpha<1$

Abstract

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel $\phi(x) = |x|^{-(1+\alpha)}$. Following our works \cite{ST2017a,ST2017b} which focused on the range $1\leq \alpha <2$, and Do et. al. \cite{DKRT2017} which covered the range $0<\alpha<1$, 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 $(\rho_0,u_0) \in H^{2+\alpha}\times H^3$, the solution approaches exponentially fast to a flocking state solution consisting of a wave $\bar{\rho}=\rho_\infty(x-t\bar{u}))$ traveling with a constant velocity determined by the conserved average velocity $\bar{u}$. The convergence is accompanied by exponential decay of all higher order derivatives of $u$.