The Navier‐Stokes equations with regularization

We consider the initial boundary value problem for the nonstationary Navier‐Stokes equations in a bounded three dimensional domain Ω with a sufficiently smooth compact boundary ∂Ω. These equations describe the motion of a viscous incompressible fluid contained in Ω for 0 < t < T and represent a system of nonlinear partial differential equations concerning four unknown functions: 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 regularized Navier‐Stokes system, which can be solved globally in time. Our construction is based on a coupling of the Lagrangian and the Eulerian representation of the fluid flow. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)