On anisotropic energy conservation criteria of incompressible fluids

In this paper, by means of divergence-free condition, we establish an anisotropic energy conservation class enabling one component of velocity in the largest space $L^3(0,T;B_{3,\mathrm{\infty }}^{1\mathrm{/}3})$ for the 3D inviscid incompressible fluids, which extends the celebrated result obtained by Cheskidov, Constantin, Friedlander and Shvydkoy in [15, Nonlinearity 21 (2008)]. For viscous flows, we generalize famous Lions’s energy conservation criteria to allow the horizontal components and vertical part of velocity to have different integrability.