Issue |
ESAIM: COCV
Volume 30, 2024
|
|
---|---|---|
Article Number | 21 | |
Number of page(s) | 36 | |
DOI | https://doi.org/10.1051/cocv/2024007 | |
Published online | 04 April 2024 |
Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D
1
Dipartimento di Matematica, Politecnico di Milano,
via Bonardi 9,
20133
Milano,
Italy
2
Dipartimento di Matematica, Politecnico di Milano,
via Bonardi 9,
20133
Milano,
Italy
* Corresponding author: andrea.poiatti@polimi.it
Received:
26
April
2023
Accepted:
15
January
2024
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.
Mathematics Subject Classification: 35K55 / 35K61 / 49J20 / 49J50 / 49K20
Key words: Convective nonlocal Cahn-Hilliard equation / Flory—Huggins potential / separation property / regularity results / optimal velocity control
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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