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, $\mathsf{W}$, representing the work done by the filter's associated sensitivity variable on the enstrophy, around which our entire analysis is centered. We show that $\mathsf{W}$ 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 $\mathsf{W}$ exceeds a certain threshold, in which case the candidate viscosity obeys an explicit error estimate that depends inversely on $\mathsf{W}$ and the observational density $N$, but directly on the error between initial conditions, or else $\mathsf{W}$ 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.