Stability estimate for scalar image velocimetry

In this paper we analyse the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors ${u}$ from images of a passive scalar field $\psi$ by minimising a cost functional, that penalises the difference between the reconstructed scalar field $\phi$ and the measured scalar field $\psi$, under the constraint that $\phi$ is advected by the reconstructed velocity field ${u}$, which again is governed by the Navier-Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence, that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two dimensional case, smooth solutions of the Navier-Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field $\psi$ to the reconstructed velocity field $u$, on any interior subset, is H\"older continuous.