Excitation and Identifiability in the 2D Navier-Stokes equations
Abstract
Convergence in parameter estimation classically requires ``persistency of excitation," which is a non-degeneracy condition on a trajectory-dependent signal. This paper develops, to the best of our knowledge, the first such excitation theory for a nonlinear partial differential equation. In the context of identifying the unknown viscosity from spectral observations in the two-dimensional Navier--Stokes equations for incompressible fluids, our excitation condition is computable, verifiable a priori, and sharp with respect to scaling. Our approach employs a data assimilation methodology to account for an unknown initial state and select candidate viscosities by minimizing the loss between the low-mode observations of the fluid velocity and low-mode projection of a nudging-based filter that assimilates these observations. The main novelty of our framework is to distinguish a new functional, W, representing the work done by the filter's associated sensitivity variable on the enstrophy, around which our entire analysis is centered. We show that W is explicitly comparable to the observability Gramian of the associated Gauss--Newton iteration, and subsequently establish the following dichotomy at every critical point of the observational loss: either W exceeds a certain threshold, in which case the candidate viscosity obeys an explicit error estimate that depends inversely on W and the observational density N, but directly on the error between initial conditions, or else W is below the threshold and the observations are quantitatively insensitive to parameter updates in a way that is detectable to the user. Notably, our approach is energy-based, and therefore expected to be adaptable to many other nonlinear dissipative systems.
Related
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- Wasserstein regularity of vorticity solutions to the 2D Navier-Stokes equations
BibTeX
@misc{martinez2026excitation,
title = {{Excitation and Identifiability in the 2D Navier-Stokes equations}},
author = {Vincent R. Martinez and Sarah Strikwerda and Xiang Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31230},
eprint = {2609.31230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31230},
}
RIS
TY - UNPB
TI - Excitation and Identifiability in the 2D Navier-Stokes equations
AU - Vincent R. Martinez
AU - Sarah Strikwerda
AU - Xiang Wan
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31230
UR - https://arxiv.org/abs/2609.31230
AB - Convergence in parameter estimation classically requires ``persistency of excitation," which is a non-degeneracy condition on a trajectory-dependent signal. This paper develops, to the best of our knowledge, the first such excitation theory for a nonlinear partial differential equation. In the context of identifying the unknown viscosity from spectral observations in the two-dimensional Navier--Stokes equations for incompressible fluids, our excitation condition is computable, verifiable a priori, and sharp with respect to scaling. Our approach employs a data assimilation methodology to account for an unknown initial state and select candidate viscosities by minimizing the loss between the low-mode observations of the fluid velocity and low-mode projection of a nudging-based filter that assimilates these observations. The main novelty of our framework is to distinguish a new functional, W, representing the work done by the filter's associated sensitivity variable on the enstrophy, around which our entire analysis is centered. We show that W is explicitly comparable to the observability Gramian of the associated Gauss--Newton iteration, and subsequently establish the following dichotomy at every critical point of the observational loss: either W exceeds a certain threshold, in which case the candidate viscosity obeys an explicit error estimate that depends inversely on W and the observational density N, but directly on the error between initial conditions, or else W is below the threshold and the observations are quantitatively insensitive to parameter updates in a way that is detectable to the user. Notably, our approach is energy-based, and therefore expected to be adaptable to many other nonlinear dissipative systems.
ER -