In this paper, we study the finite element method of the Navier–Stokes equations with the initial data belonging to theL2space for all timet>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in theH1-norm, whent∈[0,1). 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 inH1-norm and the pressure inL2-norm.