Optimal Control for Suppression of Singularity in Chemotaxis via Flow Advection

This work focuses on the optimal control design for suppressing the singularity formation in chemotaxis governed by the parabolic-elliptic Patlak–Keller–Segel (PKS) system via flow advection. The main idea of this work lies in utilizing flow advection for enhancing diffusion as to control the nonlinear behavior of the system. The objective is to determine an optimal strategy for adjusting flow strength so that the possible finite time blow-up of the solution can be suppressed. Rigorous proof of the existence of an optimal solution and derivation of first-order optimality conditions for solving such a solution are presented. Spline collocation methods are employed for solving the optimality conditions. Numerical experiments based on 2D cellular flows in a rectangular domain are conducted to demonstrate our ideas and designs.