The Navier‐Stokes equations with particle methods

Abstract

AbstractWe consider the initial boundary value problem for the nonstationary Navier‐Stokes equations in a bounded three‐dimensional domain Ω with a sufficiently smooth boundary ∂Ω. These equations describe the motion of a viscous incompressible fluid contained in Ω for 0 < t < T. They represent a system of nonlinear partial differential equations concerning four unknown functions, i.e. the velocity vector v = (v1(t, x), v2(t, x), v3(t, x)) and the kinematic pressure function p = p(t, x) of the fluid at time t ∈ (0, T) in x ∈ Ω. The purpose of this paper is to construct a Leray‐Hopf type weak solution to the nonstationary Navier‐Stokes system, which exists globally in time. Our construction is based on a suitable approximation using particle methods for stationary fluid flow. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)