ON THE ENERGY EQUALITY OF THE NAVIER – STOKES EQUATIONS IN BOUNDED THREE DIMENSIONAL DOMAINS

Abstract

The energy equality

1

‖ ( )‖ +

‖∇ ‖

 =

1

‖  ‖

2

2

is an open problem for the Navier-Stokes equations. In this paper we present a condition for the energy equality of weak solutions to the Navier – Stokes equations in bounded three dimensional domains. We prove that the energy equality holds for weak solutions in the functional class

(0,  ;        )        ≥       .