Global existence and stability of nearly aligned flocks

Abstract

We study regularity of a hydrodynamic singular model of collective behavior introduced in \cite{ST1}. In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data $(u,\rho )$ with small velocity variations $|u(x) - u(y)| < \e$ relative to its higher order norms, gives rise to a unique global regular solution which aligns and flocks exponentially fast. Moreover, we prove that the limiting flocks are stable.

The result extended to full range $0<\a<2$, proof of flocking and stability added

References in corpus (3)

• Global regularity for the fractional Euler alignment system
• Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0<\alpha <1$
• Eulerian dynamics with a commutator forcing

Cited by in corpus (9)

• Singular Cucker-Smale Dynamics
• Flocking hydrodynamics with external potentials
• On the Euler-Alignment system with weakly singular communication weights
• Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems
• Flocking with short-range interactions
• Density-induced Consensus Protocol
• Weak and Strong Solutions to the Forced Fractional Euler Alignment System
• Eulerian dynamics in multi-dimensions with radial symmetry
• Global regularity for a 1D Euler-alignment system with misalignment