Stability estimate for scalar image velocimetry

Abstract

In this paper, we analyze 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 {{u}} from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field φ and the measured scalar field ψ, under the constraint that φ is advected by the reconstructed velocity field u {{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 ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous.