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

[摘要] 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.

[发布机构] University of Alberta

[效力级别] Ferrofluid