Continuous data assimilation for a class of non-Newtonian fluids of differential type in 2d and 3d

This study addresses a class of non-Newtonian fluids governed by third-grade fluid equations, which exhibit complex and highly non-linear dynamics. To enhance the model's accuracy and state prediction in both two- and three- dimensional settings, particularly in the absence of initial conditions, we employ continuous data assimilation techniques. By incorporating velocity observational measurements - via general interpolant operators - into fluid dynamics, we derive an approximation algorithm. We first prove that the latter is well-posed, and then demonstrate that it ensures exponential stabilization of the state model as time goes to ∞. As a consequence, we show that the reconstructed state predicts the true solution within a prescribed level of accuracy over a finite time interval.