The Navier‐Stokes equations with particle methods

We 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  = ( v 1 ( t, x ), v 2 ( t, x ), v 3 ( 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)