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 $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^{1/2}$
,
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 the general case of third grade, complicates substantially the
proof of the existence of the solution. We also prove further
regularity results by a method similar to
that of Cioranescu and Girault for second grade fluids. This
extension to the third grade fluids is not straightforward, because of a
transport equation which is much more complex.