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 $α\in(0,2)$. The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range $1<α<2$ and establish the global regularity of smooth solutions, together with a full description of their long time dynamics. Specifically, we develop the flocking theory of these solutions and show long time convergence to traveling wave with rapidly aligned velocity field.
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 $α\in(0,2)$. 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 exte…
We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.
We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in \cite{STtopo}. While these systems exhibit flocking behavior emerging from purely local communication, bearing direct relevance to empirical field studies, global and even local well-posedness has proved to be a major challenge in multi-dimensional settings due to the presence of topological effects. In this paper we reveal two important classes of global smooth solutions -- parallel shear flocks with incompressible velocity and stationary density prof…
We consider the compressible Euler system with a family of nonlinear velocity alignments. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.
We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $ϕ$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + ϕ*ρ_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements abo…
In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker--Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We…
Lagrangian formulation and Eulerian closure in alignment dynamics
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Mathematics, Yonsei University（延世大学数学系）；Department of Mathematics and IPST, University of Maryland（马里兰大学数学与物理科学与技术学院）
