This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where $|{\alpha }_1+{\alpha }_2{|\le \left(24\nu \beta \right)}^{1/2}$, studied by Amrouche and Cioranescu, the H1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to...