In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 53<p<115 provided that the velocity field belongs to the critical space L2p−1(0,T;BMO(R3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO-norm. In particular, when 2<p<115, we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖u‖BMO and ‖|Du|p−22∇2u‖2.