Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910969988382720 |
|---|---|
| author | Grasselli, Maurizio Scarpa, Luca Signori, Andrea |
| author_facet | Grasselli, Maurizio Scarpa, Luca Signori, Andrea |
| contents | We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum Grasselli, Maurizio Scarpa, Luca Signori, Andrea Analysis of PDEs 35D30, 35K35, 35K86, 35Q92, 92C17 We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework. |
| title | Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum |
| topic | Analysis of PDEs 35D30, 35K35, 35K86, 35Q92, 92C17 |
| url | https://arxiv.org/abs/2404.12113 |