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The Navier‐Stokes equations with particle methods
Author(s) -
Asanalieva Nazgul,
Heutling Carolin,
Varnhorn Werner
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410352
Subject(s) - bounded function , mathematics , navier–stokes equations , domain (mathematical analysis) , compressibility , mathematical analysis , nonlinear system , boundary value problem , partial differential equation , boundary (topology) , function (biology) , flow (mathematics) , physics , geometry , mechanics , quantum mechanics , evolutionary biology , biology
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)