Pressureless Euler alignment system with control
For researchers in hydrodynamic systems and control theory, this work provides theoretical and numerical insights into controlled alignment dynamics, though it is incremental in nature.
The paper characterizes control dynamics for the pressureless Euler alignment system, identifies critical thresholds for global regularity or finite-time blow-up in 1D and 2D, and proposes a finite volume scheme with numerical validation.
We study a non-local hydrodynamic system with control. First we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that leading to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally we propose a finite volume scheme for numerical solutions of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.