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Probabilistic Methods for the Incompressible Navier–Stokes Equations With Space Periodic Conditions

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Stochastic analysis Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  04 January 2016

G. N. Milstein  and
M. V. Tretyakov
G. N. Milstein*
Affiliation:
Ural Federal University
M. V. Tretyakov*
Affiliation:
University of Leicester and University of Nottingham
*
∗ Postal address: Ural Federal University, Lenin Str. 51, 620083 Ekaterinburg, Russia. Email address: grigori.milstein@usu.ru
∗∗ Postal address: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK. Email address: michael.tretyakov@nottingham.ac.uk
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Abstract

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We propose and study a number of layer methods for Navier‒Stokes equations (NSEs) with spatial periodic boundary conditions. The methods are constructed using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the layer methods are nevertheless deterministic.

Keywords

Probabilistic representations of solutions of partial differential equationsFeynman‒Kac formulaweak approximation of stochastic differential equationslayer methodHelmholtz‒Hodge decomposition

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 65M25: Method of characteristics 65M12: Stability and convergence of numerical methods 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 742 - 772
Copyright
© Applied Probability Trust 

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