Optimal distributed control for a Cahn–Hilliard–Darcy system with mass sources, unmatched viscosities and singular potential

¹ Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milano, Italy
² Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR, 27100 Pavia, Italy
³ Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy
⁴ School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, 200433 Shanghai, PR China

^* Corresponding author: haowufd@fudan.edu.cn

Received: 31 July 2023
Accepted: 1 May 2024

Abstract

We study a Cahn–Hilliard–Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the (volume) averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function φ and the chemical potential μ. The source term in the convective Cahn–Hilliard equation contains a control R that can be thought, for instance, as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with φ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S : R ↦ φ. Then we obtain the existence of an optimal control, the Fréchet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.