Atkins, Zoe (2012) Almost sharp fronts : limit equations for a two-dimensional model with fractional derivatives. PhD thesis, University of Warwick.

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Abstract

We consider the evolution of sharp fronts and almost-sharp fronts for the
↵-equation, where for an active scalar q the corresponding velocity is defined by
u = r?(−#)−(2 − ↵)/2q for 0 < ↵ < 1. This system is introduced as a model
interpolating between the two-dimensional Euler equation (↵ = 0) and the surface
quasi-geostrophic (SQG) equation (↵ = 1).
The study of such fronts for the SQG equation was introduced as a natural
extension when searching for potential singularities for the three-dimensional Euler
equation due to similarities between these two systems, with sharp-fronts corresponding
to vortex-lines in the Euler case (Constantin et al., 1994b).
Almost-sharp fronts were introduced in C´ordoba et al. (2004) as a regularisation
of a sharp front with thickness $, with interest in the study of such solutions
as $ ! 0, in particular those that maintain their structure up to a time independent
of $. The construction of almost-sharp front solutions to the SQG equation is
the subject of current work (Fe↵erman and Rodrigo, 2012). The existence of exact
solutions remains an open problem.
For the ↵-equation we prove analogues of several known theorems for the
SQG equations and extend these to investigate the construction of almost-sharp
front solutions. Using a version of the Abstract Cauchy Kovalevskaya theorem (Safonov,
1995) we show for fixed 0 < ↵ < 1, under analytic assumptions, the existence
and uniqueness of approximate solutions and exact solutions for short-time independent
of $; such solutions take a form asymptotic to almost-sharp fronts. Finally,
we obtain the existence and uniqueness of analytic almost-sharp front solutions.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Fractional calculus, Fronts (Meteorology) -- Mathematical models
Official Date:	September 2012
Dates:	Date Event September 2012 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rodrigo, Jose L.
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	vii, 143 leaves.
Language:	eng

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