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

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.