Critical thresholds in flocking hydrodynamics with non-local alignment

Area of Science:

• Mathematical Physics
• Fluid Dynamics
• Collective Behavior

Background:

• Eulerian systems with non-local alignment model self-organized dynamics in agent-based systems.
• Existing models like Cucker-Smale and Motsch-Tadmor demonstrate flocking behavior.
• Understanding the large-time behavior and solution existence is crucial for these systems.

Purpose of the Study:

• To analyze the large-time behavior of Eulerian systems augmented with non-local alignment.
• To investigate the existence of strong solutions and their global regularity.
• To identify critical thresholds influencing solution behavior, including finite-time blow-up.

Main Methods:

• Analysis of Eulerian systems with non-local alignment interactions.
• Proof of self-organization into macroscopic flocks, analogous to agent-based models.
• Demonstration of global regularity for subcritical initial data in 1D and 2D.
• Investigation of regularity in the presence of vacuum.

Main Results:

• Non-local alignment drives strong solutions to self-organize into macroscopic flocks.
• Global regularity is proven for subcritical initial data.
• Critical thresholds in initial configurations determine global regularity or finite-time blow-up.
• The study explores the impact of vacuum on non-local alignment regularity.

Conclusions:

• The presence of non-local alignment is key to achieving organized macroscopic behavior (flocking) in Eulerian systems.
• The existence and regularity of strong solutions depend critically on initial conditions, with specific thresholds leading to blow-up.
• This research provides a theoretical framework for understanding collective dynamics and potential instabilities in such systems.