Eulerian dynamics with a commutator forcing Ⅱ: Flocking

Abstract

We continue our study of one-dimensional class of Euler equations, introduced in [ST2016], driven by a forcing with a commutator structure of the form [L φ , u](ρ) = φ * (ρu) − (φ * ρ)u, where u is the velocity field and φ belongs to a rather general class of influence or interaction kernels.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 L φ = −(−∂ xx ) α/2 . Specifically, we prove fast velocity alignment as the velocity u(·, t) approaches a constant state, u →ū, with exponentially decaying slope and curvature bounds |u x (·, t)| ∞ +|u xx (·, t)| ∞ e −δt . The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state ρ(·, t) − ρ ∞ (x −ūt) → 0. R φ(|x − y|)(f (y) − f (x))dy.

Flocking With Short-Range Interactions

“…We specifically address the case of short-range interactions D φ ≪ D S 0 . Moreover, since we do not impose any boundedness of φ, (1.8) includes both -bounded communication kernels, [4,5,12,11,2,17], and singular ones [19,20,21,23,24,25,7].…”

“…The density in (1.2) plays a central role as the carrier of local averaging, and hence a uniform bound on the density ρ(t, ·) away from vacuum is essential for the existence of strong solutions to (1.2) and their asymptotic behavior. This is particularly relevant in the case of singular interaction kernels, [23,24,25,7,26]. As noted in [26], the lower bound ρ 1 / √ 1+t will suffice to yield unconditional flocking in the general multiD case.…”

“…As noted above, theorem 1.1 does not rule out vacuum in small balls (with scale smaller than r /100); in fact, all that we evolve are averaged masses. It is known, however, that such uniform bound is intimatley related to the existence of strong (smooth) solution; see e.g., [28,24], and earlier results for non-uniform bounds in [7,23]. Here we show that our arguments can be adapted to obtain such a uniform bound of the one-dimensional density ρ away from 0 in Ω = T. We extend the techniques used in [24] which cannot be directly used with short range interaction kernels, proving in section 5 the following.…”

On Cucker–Smale dynamical systems with degenerate communication

“…The significance of condition (3) lies in the fact that such kernels prevent collisions between agents and consequently, the discrete system (1) is well-posed even though the right hand side is not Lipschitz. This issue has received extensive treatment in works of Peszek et al [2,13,14,15], and [21,22,20,6] for the Euler-alignment system. A quantitative expression of non-collision is an integral part of our approach, so we will revisit the question in Section 2.1 below.…”

“…All our results pertaining to (2) hold for a given strong solution. We note, however, that in a variety of situations, both for smooth and singular kernels, such solutions have been constructed, see [1,6,21,22,20,23,24]. In particular, the 1D case is completely understood, and small initial data results are known in multi-D case, [5,7,9,23,19].…”