Eulerian dynamics with a commutator forcing II: flocking

Abstract

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 $[\aL_ϕ,u](ρ)=ϕ*(ρu)- (ϕ*ρ)u$, 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 $ϕ(r) = r^{-(1+\a)}$ of order $\alpha \in [1,2)$ associated with the action of the fractional Laplacian $\aL_ϕ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(\cdot ,t)$ approaches a constant state, $u\to \bar{u}$, with exponentially decaying slope and curvature bounds $|u_x(\cdot,t)|_{\infty}+ |u_{xx}(\cdot,t)|_{\infty}\lesssim e^{-\d t}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $\rho (\cdot ,t)-{\rho }_{\infty }(x-\bar{u}t)\to 0$.

References in corpus (1)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Cited by in corpus (1)

• Global regularity of two-dimensional flocking hydrodynamics