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arXiv · 2304.12074

Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D

Andrea Poiatti·Andrea Signori
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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 H\"older 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.

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Andrea Poiatti, Andrea Signori. 2023-04-24. Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D. https://arxiv.org/abs/2304.12074

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