Boundary control for optimal mixing by two-dimensional second-grade fluids
Abstract
We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal (H1(Ω))' mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both g and ∂tg enter the state equation, where g is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient β and enstrophy reward weight ζ (that is, β=ζ=0), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties γ.
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BibTeX
@misc{kinra2026boundary,
title = {{Boundary control for optimal mixing by two-dimensional second-grade fluids}},
author = {Kush Kinra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09946},
eprint = {2609.09946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09946},
}
RIS
TY - UNPB
TI - Boundary control for optimal mixing by two-dimensional second-grade fluids
AU - Kush Kinra
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09946
UR - https://arxiv.org/abs/2609.09946
AB - We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal (H^1(Ω))' mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both g and ∂_tg enter the state equation, where g is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient β and enstrophy reward weight ζ (that is, β=ζ=0), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties γ.
ER -