Stability estimate for scalar image velocimetry
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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 ψ by minimising a cost functional, that penalises 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, 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.
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