[Paper Review] Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D

Sergio Frigeri, Elisabetta Rocca+1 more|arXiv (Cornell University)|Nov 6, 2014

Solidification and crystal growth phenomena22 references16 citations

TL;DR

This paper establishes the existence of optimal distributed controls and derives first-order necessary optimality conditions for a nonlocal Cahn–Hilliard/Navier–Stokes system in 2D, modeling phase separation in incompressible immiscible fluids with nonlocal interactions. The analysis builds on recent well-posedness results and uses adjoint systems and variational inequalities to characterize optimal controls via projection formulas.

ABSTRACT

We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier--Stokes system with a convective nonlocal Cahn--Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

Motivation & Objective

• To establish the existence of optimal distributed controls for a nonlocal Cahn–Hilliard/Navier–Stokes system in two spatial dimensions.
• To derive first-order necessary optimality conditions for a distributed control problem involving the fluid velocity and phase variable.
• To extend the analysis of optimal control to nonlocal models with convolution-type interactions, which are physically relevant for phase separation with long-range molecular forces.
• To provide a rigorous analytical framework for optimal control of diffuse interface models with nonlocal Cahn–Hilliard equations and $oldsymbol{ u}$-dependent viscosity.
• To fill a gap in the literature by offering a full, non-time-discretized analysis of optimal control for this nonlocal system, unlike prior works that relied on time-discretization or numerical schemes.

Proposed method

• The study employs a nonlocal Cahn–Hilliard/Navier–Stokes system coupling the Navier–Stokes equations with a convective, nonlocal Cahn–Hilliard equation in a bounded 2D domain.
• The control is applied as a distributed force $oldsymbol{v}$ in the momentum equation, acting on the fluid velocity field.
• The analysis relies on recently established well-posedness and regularity results for the state system, ensuring existence and uniqueness of strong solutions.
• An adjoint system is derived by formal Lagrangian techniques, involving dual variables $oldsymbol{p}$ and $q$ for the velocity and phase variable, respectively.
• Variational inequalities are used to characterize the optimal control, leading to a projection formula in $L^2(Q)^2$ involving the adjoint state and the control bounds.
• The derivation involves integration by parts, symmetry of the kernel $K$, and careful handling of boundary conditions and weak formulations in appropriate function spaces.

Experimental results

Research questions

• RQ1 Does an optimal distributed control exist for the nonlocal Cahn–Hilliard/Navier–Stokes system in 2D with $oldsymbol{ u}$-dependent viscosity?
• RQ2 What are the first-order necessary optimality conditions for this optimal control problem?
• RQ3 How can the optimal control be characterized in terms of the adjoint state and control constraints?
• RQ4 Can the nonlocal nature of the Cahn–Hilliard equation be rigorously incorporated into the optimal control framework without time discretization?
• RQ5 Is the optimal control characterized by a projection formula onto the admissible control set, and what does this imply for numerical implementation?

Key findings

• The paper proves the existence of at least one optimal control in the admissible set $oldsymbol{v} o oldsymbol{v} ext{ in } L^2(Q)^2$ for the nonlocal Cahn–Hilliard/Navier–Stokes system in 2D.
• First-order necessary optimality conditions are derived in the form of a variational inequality involving the adjoint state $oldsymbol{p}$ and the control difference $oldsymbol{v} - oldsymbol{ar{v}}$.
• The optimal control $oldsymbol{ar{v}}$ is characterized by a projection formula: $oldsymbol{ar{v}} = ext{Proj}_{oldsymbol{ u}_{ad}}ig(-rac{1}{eta}oldsymbol{p}ig)$, where $eta$ is a positive parameter.
• For $eta > 0$, the optimal control satisfies a pointwise projection condition: $ar{v}_i(x,t) = ext{max}igrace{v_{a,i}(x,t), ext{min}igrace{-eta^{-1}p_i(x,t), v_{b,i}(x,t)}igraceigrace$ a.e. in $Q$.
• The analysis rigorously handles the nonlocal term $K * ho$ via symmetry of the kernel $K$, which allows for symmetric integration by parts in the adjoint derivation.
• The results are valid for a regular double-well potential $F$, and the viscosity $ u( ho)$ is allowed to depend on the phase variable $ ho$, enhancing physical realism.