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

TL;DR

This paper analyzes the long-term behavior of a 3D Brinkman-Forchheimer-extended Darcy model with unknown damping parameters, using continuous data assimilation to estimate the solution and parameters over time.

Contribution

It extends the analysis of data assimilation techniques to a complex 3D nonlinear flow model with unknown parameters, providing new estimates for long-time error behavior.

Findings

Estimates in L2 and H1 norms for long-time error between true and assimilated solutions.

Demonstrates the effectiveness of continuous data assimilation in estimating unknown damping parameters.

Provides theoretical bounds on the error growth over time.

Abstract

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 L2 and H1 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.