Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2023
- DOI:
- 10.48550/arXiv.2304.12074
- arXiv:
- arXiv:2304.12074
- Bibcode:
- 2023arXiv230412074P
- Keywords:
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- Mathematics - Optimization and Control;
- Mathematics - Analysis of PDEs