On a class of conserved phase field systems with a maximal monotone perturbation

TL;DR

This paper establishes existence, regularity, and uniqueness of solutions for a perturbed Cahn-Hilliard phase field system with maximal monotone operators, and explores a sliding mode control law for temperature and phase regulation.

Contribution

It introduces a novel analysis for a phase field system with maximal monotone perturbation and applies sliding mode control to regulate temperature and phase.

Findings

Proved existence and regularity of solutions.

Established continuous dependence and uniqueness.

Demonstrated finite-time convergence under sliding mode control.

Abstract

We prove existence and regularity for the solutions to a Cahn-Hilliard system describing the phenomenon of phase separation for a material contained in a bounded and regular domain. Since the first equation of the system is perturbed by the presence of an additional maximal monotone operator, we show our results using suitable regularization of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments. Next, under further assumptions, we deduce a continuous dependence estimate whence the uniqueness property is also achieved. Then, we consider the relating sliding mode control (SMC) problem and show that the chosen SMC law forces a suitable linear combination of the temperature and the phase to reach a given (space dependent) value within finite time.