ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

In this paper, we first introduce the concept of absolutely continuous functions of order s ( 0 < s <= 1). Next, we prove the energy equality for weak solutions of the Navier-Stokes equations (NSE) in bounded three dimensional domains if and only if u is an absolutely continuous solution of order 1/2. Finally, we present a sufficient condition for the energy equality of weak solutions to NSE. Here, we prove that if u is an element of L-2 (0,T,H-s) boolean AND L-4 (0, T; L (12/2s+1) ) ( <= s < 5/2), then the energy equality holds.