Parameter Recovery and Sensitivity Analysis for the 2D Navier-Stokes Equations Via Continuous Data Assimilation

Abstract

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

39 pages, 5 figures, 1 table

References in corpus (8)

• Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone
• Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model
• Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field
• Accuracy and Stability of The Continuous-Time 3DVAR Filter for The Navier-Stokes Equation
• Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations
• Inferring flow parameters and turbulent configuration with physics-informed data-assimilation and spectral nudging
• Nonlinear Continuous Data Assimilation
• Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm