Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations

Clark, E., Katzourakis, N. and Muha, B. (2021) Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations. Nonlinearity, 35 (1). pp. 470-491. ISSN 1361-6544 doi: 10.1088/1361-6544/ac372a

Abstract/Summary

We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier–Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler–Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence-form counterpart of the corresponding non-divergence Aronsson–Euler system.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/101801
Identification Number/DOI	10.1088/1361-6544/ac372a
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Uncontrolled Keywords	Paper, Navier–Stokes equations, calculus of variations in L ∞, PDE-constrained optimisation, Euler–Lagrange equations, Aronsson–Euler systems, data assimilation, 35Q30, 35D35, 35A15, 49J40, 49K20, 49K35
Publisher	IOP Publishing
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