Uniform Error Estimates of the Finite Element Method for the Navier–Stokes Equations in [Formula: see text] with L(2) Initial Data

In this paper, we study the finite element method of the Navier–Stokes equations with the initial data belonging to the [Formula: see text] space for all time [Formula: see text]. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the [Formula: see text]-norm, when [Formula: see text]. Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in [Formula: see text]-norm and the pressure in [Formula: see text]-norm.