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Local Well-Posedness and Prodi-Serrin type Conditions for models of Ferrohydrodynamics

  • Author / Creator
    Farr, Quinton R
  • Ferrofluids are liquids with a magnetic colloid suspended by Brownian motion which become magnetized in the presence of an external magnetic field. They have recently garnered a lot of interest due to their wide range of applications in industry and biomedicine. For example, ferrofluids are used in rotary seals, microchannel flows, nanotechnology, and even in experimental cancer treatments. There are two widely-used mathematical models which describe the motion of ferrofluids; the Rosensweig model derived by Ronald E. Rosensweig, and the Shliomis model derived by Mark I. Shliomis. In the mathematical literature, the ferrofluid models remain relatively unexplored. Only a handful of papers have been written on them, most of which are concerned with existence of weak solutions or strong solutions. Because the equations describing ferrofluids build upon the famous Navier-Stokes equations, we expect many properties which have been proven in the far more extensive literature for those equations to have an analogous version of themselves hold for the ferrofluid models. This thesis helps to narrow this gap in the literature by extending the analysis of classical solutions to both models. In particular, we first show local well-posedness of classical solutions on the whole three dimensional Euclidean space for a regularized version of each model-- that a solution exists, is unique in this class of solutions, and varies continuously with the initial data in the appropriate topology. Then, we derive so-called Prodi-Serrin type conditions for the solutions, which are sufficient conditions to extend the solutions we constructed up to and beyond a time T.

  • Subjects / Keywords
  • Graduation date
    2016-06:Fall 2016
  • Type of Item
    Thesis
  • Degree
    Master of Science
  • DOI
    https://doi.org/10.7939/R3DR2PK76
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Master's
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Applied Mathematics
  • Supervisor / co-supervisor and their department(s)
    • Xinwei Yu (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Peter Minev (Mathematical and Statistical Sciences)
    • Rouslan Krechetnikov (Mathematical and Statistical Sciences)
    • Thomas Hillen (Mathematical and Statistical Sciences)