Nonlocal Cahn-Hilliard-Brinkman System with Regular Potential: Regularity and Optimal Control

Abstract

In this paper, we study an optimal control problem for nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. The system evolves with regular potential in a two-dimensional bounded domain. We extend the existence of weak solution results for the system to prove the existence of strong solution under extra assumptions on the forcing term and initial datum. Further, using our regularity results, we study the tracking type optimal control problem. We prove the existence of optimal control and establish the first-order optimality condition. Lastly, we characterise optimal control in terms of the solution of the corresponding adjoint system. The existence of the solution for the adjoint system is also established.