Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Abstract

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the $L^2$ and $H^1$ norms of error are bounded by a constant times a power of the Voigt-regularization parameter $\alpha >0$, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as $\alpha$ goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the $H^2$ norm.

References in corpus (9)

• The Camassa-Holm equations as a closure model for turbulent channel flow
• Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone
• Discrete Data Assimilation in the Lorenz and 2D Navier--Stokes Equations
• Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field
• 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
• Global in time stability and accuracy of IMEX-FEM data assimilation schemes for the Navier-Stokes equations
• Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm
• Self-truncation and scaling in Euler-Voigt-$\alpha$ and related fluid models