The third‐grade fluid equations are a type of Navier–Stokes equation with perturbations, where the perturbation term is given by a third‐grade fluid. This article primarily investigates the large‐time behavior for a class of third‐grade fluid equations. Utilizing the classical Fourier decomposition approach and energy methods, we first study the weak solutions of the equation, focusing on algebraic decay under the initial velocity condition. At the same time, we further explore the error estimates between the third‐grade fluid equation and Navier–Stokes equation. Our research is of significant importance for a deeper understanding of the asymptotic evolution of viscous incompressible fluids, and its results will greatly contribute to the further study and development of issues related to the Navier–Stokes equation.
Song et al. (Thu,) studied this question.