On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential

Giulio Schimperna

doi:10.3934/dcdss.2022008

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2022, Volume 15, Issue 8: 2305-2329. Doi: 10.3934/dcdss.2022008

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$ \Gamma $

-compactness and

$ \Gamma $

-stability of the flow of heat-conducting fluids

On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential

Giulio Schimperna

Dipartimento di Matematica, Università di Pavia, Via Ferrata, 5, I-27100 Pavia, Italy

Dedicated to Maurizio Grasselli on the occasion of his 60th birthday, with friendship and admiration

Received: August 31, 2021

Published: August 2022

This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018-2022), Department of Mathematics "F. Casorati", University of Pavia. The present paper also benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations" and of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica)

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Abstract

We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter $ \varphi $ is designed in such a way to keep $ \varphi $ in between the reference interval $ (-1, 1) $ despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.

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Mathematics Subject Classification: Primary: 35D30, 35K61; Secondary: 35Q35, 76D27, 92C30.

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