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Some problems in the mathematical theory of fluid mechanics
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Dashti, Masoumeh (2008) Some problems in the mathematical theory of fluid mechanics. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2338533~S15
Abstract
This thesis addresses three problems related to the mathematical theory of fluid mechanics.
Firstly, we consider the threedimensional incompressible NavierStokes equations
with an initial condition that has H1Sobolev regularity. We show that there is
an a posteriori condition that, if satisfied by the numerical solutions of the equations,
guarantees the existence of a strong solution and therefore the validity of the numerical
computations. This is an extension of a similar result proved by Chernyshenko,
Constantin, Robinson & Titi (2007) to less regular solutions not considered by them.
In the second part, we give a simple proof of uniqueness of fluid particle trajectories
corresponding to the solution of the ddimensional Navier Stokes equations, d = 2, 3,
with an initial condition that has H(d/2)−1Sobolev regularity. This result has been
proved by Chemin & Lerner (1995) using the LittlewoodPayley theory for the flow in
the whole space Rd. We provide a significantly simpler proof, based on the decay of
Sobolev norms ( of order more than (d/2)−1 ) of the velocity field after the initial time,
that is also valid for the more physically relevant case of bounded domains.
The last problem we study is the motion of a fluidrigid disk system in the whole
plane at the zero limit of the rigid body radius. We consider one rigid disk moving
with the fluid flow and show that when the radius of the disk goes to zero, the solution
of this system converges, in an appropriate sense, to the solution of the NavierStokes
equations describing the motion of only fluid in the whole plane. We then prove that
the trajectory of the centre of the disk, at the zero limit of its radius, coincides with a
fluid particle trajectory. We also show an equivalent result for the limiting motion of
a spherical tracer in R3, over a small enough time interval.
Item Type:  Thesis or Dissertation (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Fluid mechanics, NavierStokes equations  
Official Date:  September 2008  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Robinson, James C. (James Cooper), 1969  
Sponsors:  University of Warwick  
Extent:  132 p.  
Language:  eng  
URI:  http://wrap.warwick.ac.uk/id/eprint/3665 
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