The Boussinesq approximation is a set of fluids equations utilized in the atmospheric and oceanographic sciences. They may be thought of as inhomogeneous, incompressible Euler or Navier-Stokes equations, where the inhomogeneous term is a scalar quantity, typically representing density or temperature, governed by a convection-diffusion equation.
In this thesis, we prove local-in-time existence and uniqueness of an inviscid Boussinesq system. Furthermore, we show that under stronger assumptions, the local-in-time results can be extended to global-in-time existence and
uniqueness as well. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov-type. We use paradifferential calculus and properties of the Besov-type
spaces to control the growth of vorticity via an a priori estimate on the growth of density. This result is motivated by work of M. Vishik demonstrating local-in-time existence and uniqueness for 2D Euler equations in borderline Besov-type spaces, and by work of R. Danchin and M. Paicu showing the global well-posedness of the 2D Boussinesq system with initial data in critical Besov and Lp-spaces. / text
| Identifer | oai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/ETD-UT-2012-05-5143 |
| Date | 12 July 2012 |
| Creators | Glenn-Levin, Jacob Benjamin |
| Source Sets | University of Texas |
| Language | English |
| Detected Language | English |
| Type | thesis |
| Format | application/pdf |
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