Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

Abstract

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels φ(x) := |x| −(n+α) for α ∈ (0, 2). Here, we consider the critical case α = 1 and establish a couple of global existence results of smooth solutions, together with a full description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. In fact, using Duhamel's approach we get that any solution is almost Lipschitz-continuous in space. We extend the notion of weak solution for α ∈ [1, 2) and prove the existence of global Leray-Hopf solutions. Furthermore, we give an anisotropic Onsager-type criteria for the validity of the natural energy law for weak solutions of the system. Finally, we provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution depending solely on the size of the initial entropy.This system represents a multi-dimensional hydrodynamic version of the celebrated Cucker-Smale flocking model introduced in [7, 8], with a huge number of applications ranging from biology or robotics to social sciences, see [4,29,21] for recent surveys and references therein. The system (1) is designed to describe the interaction between agents governed by laws of self-organization with communication protocol encoded into the kernel φ.In many realistic situations and applications, the communication among agents takes place in local neighborhoods induced by short-range kernels. Particularly interesting is the case of singular communication kernels. Clearly, singularity at the origin strongly emphasizes local communication.For models with singular kernels given by φ(x) := |x| −(n+α) for 0 < α < 2 the operator L φ ≡ L α becomes the (negative of) classical fractional Laplacian:R n f (y) − f (x) |x − y| n+α dy Λ α := (−∆) α/2 , 0 < α < 2.

Global well-posedness for the 3D damped micropolar Bénard system with zero thermal conductivity

“…The exponential decays observed in (1.13) and (1.14) are commonly referred to as strong flocking. This result has already been established and documented in the literature, for instance in [19,22].…”

“…However, in the critical case of α = 1, the intricate structure of C 1 (u, ρ) presents significant challenges in extracting sufficient dissipation to counterbalance the nonlinear advection. To the best of our knowledge, the only available result in the literature is provided by Lear in [19], where global well-posedness is established for the specific case (1.7). For general equation (1.5), smallness assumptions are required to obtain global smooth solutions.…”

“…Our next result concerns the asymptotic flocking behavior of solutions to (1.5). This phenomenon has been extensively studied in the general context of the Euler-alignment system (1.1) (see, e.g., [19,20,22,23,24,25,32,36]). In particular, the global solution tends to exhibit certain collective behavior.…”

“…A regularity criterion has been established in [19], which is stated in (2.1) and asserts that solutions remain smooth if both ρ and u are Lipschitz continuous. In comparison, our regularity criterion (1.15) imposes a less stringent condition by requiring only Hölder continuity of u.…”

Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

“…We limit our brief discussion to the 1D setting. If φ is non-integrable at the origin, the alignment force produces nonlocal dissipation, and the solution is globally regular for all smooth initial data away from vacuum [20,40,51,52,53], with instant regularization for less regular initial data [45,42]. Another possibility is that φ is weakly singular, i.e., unbounded but integrable at the origin.…”