We construct an energy conserving material difference quotient that approximates the nonlinear convective term of the non-stationary Navier-Stokes equations in bounded three-dimensional domains with smooth boundary. Using a suitable time delay, we determine the trajectories of the fluid particles from the velocity field and vice versa, such that the resulting approximate equations can be solved for all times. A special initial construction of compatible data ensures that the corresponding solution is uniquely determined and has a high degree of spatial regularity uniformly in time and including   Finally, we can prove, moreover, that the sequence of the approximate solutions has an accumulation point satisfying the Navier-Stokes equations in a weak sense globally in time.

Navier-Stokes equations, time delay, material differences.