Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential

TL;DR

This paper develops an optimal control framework for a complex two-dimensional fluid mixture model involving nonlocal Cahn-Hilliard and Navier-Stokes equations, accommodating physically realistic degenerate mobility and singular potentials.

Contribution

It extends previous work by establishing optimal control results for a more realistic model with degenerate mobility and singular potential, using recent existence of strong solutions.

Findings

Proved existence of optimal control for the system.

Analyzed differentiability of the control-to-state map.

Derived first-order necessary optimality conditions.

Abstract

In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier--Stokes equations, nonlinearly coupled with a convective nonlocal Cahn--Hilliard equation. The system rules the evolution of the volume-averaged velocity $\uvec$ of the mixture and the (relative) concentration difference $\varphi$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map $\vvec \mapsto [\uvec,\varphi]$, and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E.~Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C.\,G.~Gal in [14].