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Optimal Control of a Navier-Stokes-Cahn-Hilliard System for Membrane-fluid Interaction

Andrea Signori, Hao Wu

TL;DR

This work develops a rigorous optimal control framework for a 2D fluid–membrane model where incompressible Navier–Stokes dynamics are coupled with a sixth-order phase-field (Cahn–Hilliard) equation describing membrane deformation. By analyzing the control-to-state map, the authors prove existence of optimal controls, establish Fréchet differentiability via a linearized system, and derive adjoint-based first-order optimality conditions, including a projection formula for the control. The analysis relies on strong well-posedness, energy estimates, and regularity results in two dimensions, enabling a gradient-based optimization approach. Overall, the paper provides a mathematically solid foundation for optimal control of complex fluid–structure interactions in membrane–fluid systems, with potential for robust numerical implementations.

Abstract

We consider an optimal control problem for a two-dimensional Navier-Stokes-Cahn-Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier-Stokes equations, which are nonlinearly coupled with a sixth-order Cahn-Hilliard type equation representing the deformation of a flexible membrane through a phase-field variable. Building on the previously established existence and uniqueness of global strong solutions for the coupled system, we introduce an external forcing term acting on the fluid as the control variable. Then we seek to minimize a tracking-type cost functional, demonstrating the existence of an optimal control and deriving the associated first-order necessary optimality conditions. A key issue is to establish sufficient regularity for solutions of the adjoint system, which is crucial for the rigorous derivation of optimality conditions in the fluid dynamic setting.

Optimal Control of a Navier-Stokes-Cahn-Hilliard System for Membrane-fluid Interaction

TL;DR

This work develops a rigorous optimal control framework for a 2D fluid–membrane model where incompressible Navier–Stokes dynamics are coupled with a sixth-order phase-field (Cahn–Hilliard) equation describing membrane deformation. By analyzing the control-to-state map, the authors prove existence of optimal controls, establish Fréchet differentiability via a linearized system, and derive adjoint-based first-order optimality conditions, including a projection formula for the control. The analysis relies on strong well-posedness, energy estimates, and regularity results in two dimensions, enabling a gradient-based optimization approach. Overall, the paper provides a mathematically solid foundation for optimal control of complex fluid–structure interactions in membrane–fluid systems, with potential for robust numerical implementations.

Abstract

We consider an optimal control problem for a two-dimensional Navier-Stokes-Cahn-Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier-Stokes equations, which are nonlinearly coupled with a sixth-order Cahn-Hilliard type equation representing the deformation of a flexible membrane through a phase-field variable. Building on the previously established existence and uniqueness of global strong solutions for the coupled system, we introduce an external forcing term acting on the fluid as the control variable. Then we seek to minimize a tracking-type cost functional, demonstrating the existence of an optimal control and deriving the associated first-order necessary optimality conditions. A key issue is to establish sufficient regularity for solutions of the adjoint system, which is crucial for the rigorous derivation of optimality conditions in the fluid dynamic setting.

Paper Structure

This paper contains 10 sections, 9 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation
  4. Assumptions and results
  5. The Control-to-State Operator and Its Properties
  6. Well-posedness and stability of the state system
  7. Differentiability of the control-to-state operator
  8. The Optimal Control Problem
  9. Existence of optimal controls
  10. First-order necessary optimality conditions

Key Result

Lemma 2.1

Let $\Omega$ be a bounded domain of class $C^2$ in $\mathbb{R}^2$. For any $\boldsymbol{f} \in {{\boldsymbol H}_\sigma}$, there exists a unique pair $(\boldsymbol{u}, p)\in {{\boldsymbol W}_\sigma}\times V_0$ such that $-\Delta \boldsymbol{u}+\nabla p=\boldsymbol{f}$ a.e. in $\Omega$, that is, $\bol where $C>0$ depends on $\Omega$ but is independent of $\boldsymbol{f}$.

Theorems & Definitions (26)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Well-posedness
  • Remark 2.5
  • Remark 2.6
  • Corollary 2.7: Continuous dependence for chemical potentials
  • Remark 2.8
  • Theorem 2.9: Differentiability of the control-to-state operator
  • Theorem 2.10: Existence of an optimal control
  • ...and 16 more