Energy Equality of the 3D Navier–Stokes Equations and Generalized Newtonian Equations

In this paper, we establish an energy conservation criterion via a combination of the velocity and the gradient of velocity for both the Cauchy and Dirichlet problems of 3D incompressible Navier–Stokes equations, which covers the classical result of Lions (Rend Semin Mat Univ Padova 30:16–23, 1960) and Shinbrot (SIAM J Math Anal 5:948–954, 1974) and recent results in Berselli and Chiodaroli (Nonlinear Anal 192:111704, 2020) and Zhang (J Math Anal Appl 480:9, 2019). The parallel result also holds for the weak solutions of the generalized Newtonian equations, which immediately entails the latest corresponding progress in Beirao da Veiga and Yang (Nonlinear Anal 185:388–402, 2019), Yang (Appl Math Lett 88:216–221, 2019) and Berselli and Chiodaroli (2020) and particularly derives several new sufficient conditions keeping energy equality.