Global Regularity for the Fractional Euler Alignment System

Abstract

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian (−∂ xx ) α/2 , α ∈ (0, 1). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all α ∈ (0, 1). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.Here, N i (t) = {j : |x i (t) − x j (t)| ≤ r}, with some r > 0 fixed, ∆θ is a uniformly distributed random variable in [−1, 1], and η > 0 is a parameter measuring the strength of the noise.

Flocking With Short-Range Interactions

“…holds for all initial data which give rise to smooth solutions. Thus, if φ is a long-range interaction kernel satisfying the 'fat tail' condition (1.5), then 'smooth solutions must flock', [27], and the existence of smooth solutions is known in Ω = T 1 , [7,23,24,25], in Ω = R 2 [14], and with small data in Ω = T d and Ω = R d [22,6].…”

“…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 the Euler-alignment system with weakly singular communication weights

“…It has been studied recently that the so called strongly singular interaction has a regularization effect, which prevents the solution from finite time singularity formations. In 1D, global regularity is obtained in [9] for s ∈ (1, 2), and in [17] for s ∈ [2, 3) through a different approach. 9,17]).…”

“…In 1D, global regularity is obtained in [9] for s ∈ (1, 2), and in [17] for s ∈ [2, 3) through a different approach. 9,17]). Consider the 1D Euler-Alignment system (5)-(6) with smooth periodic initial data (ρ 0 , G 0 ).…”