The energy equality for weak solutions to the equations of non-Newtonian fluids

In this short paper, we extend the result of Shinbrot (1974) to an incompressible fluid with shear dependent viscosity. It is shown that a weak solution to the equations of non-Newtonian fluids lying in a space L-q (0 ,T; L-p) satisfies an energy equality, where 2r/r-1 <= p <= 2r/r-2 and 1/p + 1/q <= r-1/r, if r > 2; p >= 2r/r-1 and r-1/p + 1/q = r-1/2, if 2(n+1) /n+2 < r <= 2. In particular, our result implies that the weak solution must satisfy the energy equality when r >= 3n+2/n+2, which is consistent with the known fact. (C) 2018 Elsevier Ltd. All rights reserved.