We prove that strong solutions of the Boussinesq equations in 2D and 3D can be extended as analytic functions of complex time. As a consequence we obtain the backward uniqueness of solutions.

We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by $p(x,t)$-power law for $p(x,t)\ge (3n+2)/(n+2)$, $n\ge 2,$ by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.