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

TL;DR

This paper investigates a variational problem constrained by the Navier-Stokes equations, establishing existence and convergence of minimisers in $L^p$ and $L^$ spaces, relevant for atmospheric data assimilation.

Contribution

It proves the existence of PDE-constrained minimisers for all $p$, shows convergence of $L^p$ to $L^$ minimisers, and characterizes the structure of $L^$ minimisers via divergence PDE systems.

Findings

Existence of minimisers for all $p$ in the constrained problem.

Convergence of $L^p$ minimisers to $L^$ minimisers as $p o $.

Characterization of $L^$ minimisers through divergence PDE systems.

Abstract

We study a minimisation problem in $L^p$ and $L^\infty$ 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 $L^\infty$ minimisers as $p\to\infty$. We further show that $L^p$ minimisers solve an Euler-Lagrange system. Finally, all special $L^\infty$ 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.