A note on the Prodi-Serrin conditions for the regularity of a weak solution to the Navier-Stokes equations

The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. The most interesting aspect of the result is that no compatibility condition is required to the initial data $v_0\in J^2(\text{OO})J2(\mathrm{\Omega })$.