  In this paper, we derive some new Gagliardo-Nirenberg type inequalities in
Lorentz type spaces without restrictions on the second index of Lorentz norms,
which generalize almost all known corresponding results. Our proof mainly
relies on the Bernstein inequalities in Lorentz spaces, the embedding relation
among various Lorentz type spaces, and Littlewood-Paley decomposition
techniques. In addition, we establish several novel criteria in terms of the
velocity or the gradient of the velocity in Lorentz spaces for energy
conservation of the 3D Navier-Stokes equations. Particularly, we improve the
classical Shinbrot's condition for energy balance to allow both the space-time
directions of the velocity to be in Lorentz spaces.
