On the Euler-alignment system with weakly singular communication weights

Abstract

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of complex biological systems modeling animal flocks. For such Euler-Alignment system with bounded interactions, a critical threshold phenomenon is proved in [18], where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in [9], for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar as the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

“…We also consider the case of "weakly singular" φ, which for the 1D hydrodynamic model means that φ(x) ∼ |x| −s near the origin, for some s ∈ (0, 1). Wellposedness of the model with these kernels has been studied in [25]. The latter resolves the case of subcritical and supercritical initial data, and shows that some critical initial data leads to finite-time blowup.…”

“…See for example [1,13,25] for more details. (Of these, only [13] explicitly includes the viscous regularization argument.)…”

“…If one works with (VV), one can avoid the problem of tracing the dynamics of Ω(t ) itself; most of the wellposedness literature on the full space setting uses this framework, cf. [1,10,24,25]. However, this technical simplification comes at a cost.…”

“…The latter resolves the case of subcritical and supercritical initial data, and shows that some critical initial data leads to finite-time blowup. However, the analysis of [25] does not apply to all critical initial data. Our second main result shows that a large class of critical initial data leads to global-in-time existence and thus slightly sharpens the criteria in [25].…”

“…We do not deal with the sharpest possible spaces, nor do we attempt to incorporate general locally integrable kernels. We instead refer the reader to the following references for more details: [1,8,13,25].…”