Parameter analysis in continuous data assimilation for three-dimensional Brinkman-Forchheimer-extended Darcy model (Q6585086)

The paper is focused on state estimation of the three-dimensional Brinkman-Forchheimer extended Darcy model, also called the three-dimensional Navier-Stokes equation with damping. It differs from the classical Navier-Stokes equation by the presence of an additional nonlinear drag term related to the pore dimension, shape and porosity, and given up to two unknown constant parameters characterizing the magnitude and exponent. Only some guesses about the values of these two parameters are available. All the other parameters in the mathematical model and the forcing term are assumed to be perfectly known. The spatial domain is a three-dimensional box and the boundary conditions are periodic. It is assumed that state observations are dense enough to interpolate the state (i.e., the spatial velocity field) continuously in time over the entire spatial domain. This interpolated solution is plugged in into the observer-like equation serving as a device to assimilate the data and to produce a much more accurate state estimate continuously in time. The presented idea of the data assimilation algorithm has already been employed elsewhere for two space dimensions. The authors' contribution lies in providing both \(L^2\) and \(H^1\) bounds to the evolving velocity estimation error in terms of the errors in the two parameters characterizing the drag term. The paper contains detailed proofs of all the results obtained with required rigour, but these nice theoretical results translate into practice, too. Specifically, they demonstrate that the presented data assimilation algorithm can be robust to the lack of precise knowledge of all physical parameters, including those related to the nonlinear parts of the system.