Time delay and Lagrangian approximation for Navier–Stokes flow

Abstract

Abstract We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for 0 < t ≤ T ${0 < t \le T}$ in a bounded domain Ω ⊆ ℝ 3 ${\Omega \subseteq \mathbb {R}^3}$ with sufficiently smooth boundary ∂ Ω ${\partial \Omega }$ . 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.