On the Cahn-Hilliard/Allen-Cahn equations with singular potentials

Alain Miranville; Wafa Saoud; Raafat Talhouk

doi:10.3934/dcdsb.2018308

Article Contents

2019, Volume 24, Issue 8: 3633-3651. Doi: 10.3934/dcdsb.2018308

This issue Previous Article Smoothness of density for stochastic differential equations with Markovian switching Next Article Invariance principle in the singular perturbations limit

On the Cahn-Hilliard/Allen-Cahn equations with singular potentials

1.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie, Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
2.
Xiamen University, School of Mathematical Sciences, Xiamen, Fujian, China
3.
Université Libanaise, Laboratoire de Mathématiques - EDST, Faculté des Sciences, Hadath, Liban

Received: February 2018

Revised: June 2018

Early access: November 2018

Published: August 2019

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Abstract

The purpose of this work is to prove the existence and uniqueness of the solution for a Cahn-Hilliard/Allen-Cahn system with singular potentials (and, in particular, the thermodynamically relevant logarithmic potentials). We also prove the existence of the global attractor. Finally, we show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.

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Mathematics Subject Classification: 35K51, 35B41, 35B45.

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