Eulerian dynamics with a commutator forcing II: flocking

We continue our study of one-dimensional class of Euler equations, introduced in \cite{ST2016}, driven by a forcing with a commutator structure of the form $[{\text{aL}}_{\varphi },u](\rho )=\varphi \ast (\rho u)-(\varphi \ast \rho )u$, where $u$ is the velocity field and $\varphi$ belongs to a rather general class of \emph{influence} or interaction kernels. In this paper we quantify the large-time behavior of such systems in terms of \emph{fast flocking} for two prototypical sub-classes of kernels: bounded positive $\varphi$’s, and singular $\varphi (r)=r^{-(1+\text{a})}$ of order $\alpha\in [1,2)$associatedwiththeactionofthefractionalLaplacian$\aL_\phi=-(-\partial_{xx})^{\alpha/2}$. Specifically, we prove fast velocity alignment as the velocity $u(\cdot ,t)$ approaches a constant state, $u \to \bar{u}$,withexponentiallydecayingslopeandcurvaturebounds$|u_x(\cdot,t)|{\infty}+ |u{xx}(\cdot,t)|{\infty}\lesssim e^{-\d t}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $\rho(\cdot,t) - {\rho{\infty}}(x - \bar{u} t) \rightarrow 0$.