We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for in a bounded domain with sufficiently smooth boundary . We use a particle method in connection with a time delay to approximate the nonlinear convective term by a single central Lagrangian difference quotient constructed from autonomous systems of ordinary differential equations. We show that the resulting approximate Navier–Stokes system has a uniquely determined global solution satisfying the energy equation and having a high degree of regularity uniformly in time. Moreover, we prove that the sequence of approximate solutions has an accumulation point satisfying the Navier–Stokes equations in a weak sense and the energy inequality.
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