An optimal control problem for Stokes-Cahn-Hilliard-Oono equations with regular potential

TL;DR

This paper formulates and analyzes an optimal control problem for a coupled Stokes-Cahn-Hilliard-Oono system modeling two immiscible fluids with surface tension, establishing existence and optimality conditions.

Contribution

It introduces a new optimal control framework for the Stokes-Cahn-Hilliard-Oono equations with a regular potential, including existence and adjoint-based optimality conditions.

Findings

Existence of an optimal control solution is proven.

Optimality conditions are derived via an adjoint system.

The model captures dynamic diffuse interfaces between fluids.

Abstract

This article discusses an optimal control problem for a phase field model of two immiscible incompressible fluid flow, incorporating surface tension effects. The optimal control problem is defined with a $L^2$-cost functional and subject to the constraints governed by a system of coupled Stokes-Cahn-Hilliard-Oono equations. In this model, fluids are separated by a dynamic diffuse interface of finite width. We investigate the optimality condition of a given control. Initially, we establish the existence of an optimal solution for the coupled optimal control problem. Subsequently, we derive the optimality condition with respect to the corresponding adjoint system.