Eulerian dynamics with a commutator forcing Ⅱ: Flocking

We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form $[{\mathcal L}_φ, u](ρ)$, where $u$ is the velocity field and ${\mathcal L}_φ$ belongs to a rather general class of convolution operators depending on interaction kernels $φ$.

In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive $φ$'s, and singular $φ(r) = r^{-(1+α)}$ of order $α∈ [1, 2)$ associated with the action of the fractional Laplacian ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(·, t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.