Global existence in the energy space of the solutions of a non-Newtonian fluid

Existence of global weak solutions for the equations of third-grade fluids in RdRd for d=2,3d=2,3 is investigated. The main contribution of the present paper is to show that the classical monotonicity methods can be applied to third-grade fluids in order to construct global H1(Rd)H1(Rd) weak solutions. More precisely, for initial data in H1(Rd)H1(Rd), global H1(Rd)H1(Rd) weak solutions which satisfy an energy equality are constructed. The energy equality for weak solutions is an important fact and we point out that for the Navier–Stokes equations the validity of the energy equality is still an open problem. In the two-dimensional case, it is shown that the H2(R2)H2(R2) weak solutions are unique in the class of constructed H1(R2)H1(R2) weak solutions. These results improve those of V. Busuioc and D. Iftimie who have previously proved the existence of an H2(Rd)H2(Rd) global weak solution and the uniqueness of this H2H2 solution in the two-dimensional case.