Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

Abstract

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 divergenceform counterpart of the corresponding non-divergence Aronsson-Euler system. Contents 1. Introduction and main results 1 2. Preliminaries and the main estimates 9 3. Minimisers of L p problems and convergence as p → ∞ 12 4. The equations for L p PDE-constrained minimisers 13 5. The equations for L ∞ PDE-constrained minimisers 15 Acknowledgement 19 References 19