Title page for ETD etd-06162011-055842


Document Type Doctoral Thesis
Author Deugoue, Gabriel
Email gadeugoue@yahoo.fr
URN etd-06162011-055842
Document Title Existence of solutions for stochastic Navier-Stokes alpha and Leray alpha models of fluid turbulence and their relations to the stochastic Navier-Stokes equations
Degree PhD
Department Mathematics and Applied Mathematics
Supervisor
Advisor Name Title
Prof M Sango Supervisor
Keywords
  • stochastic three dimensional Navier-Stokes-∝
  • stochastic three dimensional Leray-∝ model
Date 2011-04-05
Availability unrestricted
Abstract
In this thesis, we investigate the stochastic three dimensional Navier-Stokes-∝ model and the stochastic three dimensional Leray-∝ model which arise in the modelling of turbulent flows of fluids.

We prove the existence of probabilistic weak solutions for the stochastic three dimensional Navier-Stokes-∝ model. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. We also discuss the uniqueness. The proof of the existence combines the Galerkin approximation and the compactness method. We also study the asymptotic behavior of weak solutions to the stochastic three dimensional Navier-Stokes-∝ model as ∝ approaches zero in the case of periodic box. Our result provides a new construction of the weak solutions for the stochastic three dimensional Navier-Stokes equations as approximations by sequences of solutions of the stochastic three dimensional Navier-Stokes-∝ model.

Finally, we prove the existence and uniqueness of strong solution to the stochastic three dimensional Leray-∝ equations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation combines with the weak convergence methods. We also study the asymptotic behavior of the strong solution as alpha goes to zero. We show that a sequence of strong solution converges in appropriate topologies to weak solutions of the stochastic three dimensional Navier-Stokes equations.

All the results in this thesis are new and extend works done by several leading experts in both deterministic and stochastic models of fluid dynamics.

2010 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.

Please cite as follows:

Deugoue, G 2010, Existence of solutions for stochastic Navier-Stokes alpha and Leray alpha models of fluid turbulence and their relations to the stochastic Navier-Stokes equations, PhD thesis, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-06162011-055842/ >

D11/395/ag

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