Time delay and Lagrangian approximation for Navier–Stokes flow

Nazgul Asanalieva; Carolin Heutling; Werner Varnhorn

doi:10.1515/anly-2015-0003

Published by Oldenbourg Wissenschaftsverlag May 19, 2015

Time delay and Lagrangian approximation for Navier–Stokes flow

Nazgul Asanalieva , Carolin Heutling and Werner Varnhorn

From the journal Analysis

https://doi.org/10.1515/anly-2015-0003

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Abstract

We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for 0<t≤T in a bounded domain Ω⊆ℝ3 with sufficiently smooth boundary ∂Ω. 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.

Keywords: Navier–Stokes equations; Lagrangian approximation; time delay; weak solutions

MSC: 35B65; 35D30; 35Q30; 76D05

Received: 2015-1-5

Revised: 2015-3-9

Accepted: 2015-4-30

Published Online: 2015-5-19

Published in Print: 2015-11-1

Time delay and Lagrangian approximation for Navier–Stokes flow

Abstract

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