On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation

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 related 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 (spacedependent) value within finite time.

Sliding Mode Control for a Phase Field System Related to Tumor Growth

“…In that case it was possible to have different choices for the manifold: in particular, either one of the physical variables or a combination of them could remain constant in time. With reference to the results of [3], we aim to mention the analyses developed in [13,14]: in particular, the second contribution is devoted to a conserved phase field system with a SMC feedback law for the internal energy in the temperature equation.…”

Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics

“…Besides, we quote [10], where a first simplified version of the entropy system is considered, and [9,11] for related analyses and results. Besides, let us mention the contributions [18,19], where standard phase field systems of Caginalp type, perturbed by the presence of nonlinearities similar to (6), are considered and the existence of strong solutions, the global well-posedness of the system and the sliding mode property are proved. We also refer to [14], where the author prove the existence of solutions for a system characterized by the contemporary presence of two nonlinearities in the entropy balance equation: the resulting system is highly nonlinear and the main difficulties lie in the treatment of the doubly nonlinear equation…”

Sliding mode control for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation

“…SMCs are useful in many applications: we cite previous studies concerning the control of semilinear PDE systems and the recent contribution, where a sliding mode approach is applied for the first time to phase field systems of Caginalp type. We also mention the analysis developed in other studies: in particular, the second contribution is devoted to a conserved phase field system with a SMC feedback law for the internal energy in the temperature equation.…”