Parameter analysis in continuous data assimilation for three-dimensional Brinkman–Forchheimer-extended Darcy model

In this paper, we study analytically the long-time behavior of three-dimensional Brinkman–Forchheimer-extended Darcy model, in the context that the parameters related to the damping nonlinear term are unknown. This work is inspired by the approach firstly introduced for two-dimensional Navier–Stokes equations by Carlson, Hudson and Larios. We show estimates in \(L^2\) $L^2$ and \(H^1\) $H^1$ for large-time error between the true solution and the assimilated solution, which is constructed with the unknown damping parameters and observational measurements obtained continuously in time from a continuous data assimilation technique proposed by Azouani, Olson and Titi.