Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913199231598592 |
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| author | Poiatti, Andrea Signori, Andrea |
| author_facet | Poiatti, Andrea Signori, Andrea |
| contents | In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12074 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D Poiatti, Andrea Signori, Andrea Optimization and Control Analysis of PDEs In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established. |
| title | Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2304.12074 |