engleski

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

We study a minimisation problem in Lp 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 Lp minimisers converge to L∞ minimisers as p→∞. We further show that Lp minimisers solve an Euler- Lagrange system. Finally, all special L∞ minimisers constructed via approximation by Lp 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.

Navier–Stokes equations ; calculus of variations in L∞ ; PDE-constrained optimisation ; Euler–Lagrange equations ; Aronsson–Euler systems ; data assimilation

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