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Some problems in irregular ordinary differential equations
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Sharples, Nicholas (2012) Some problems in irregular ordinary differential equations. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2585398~S1
Abstract
We study the non-autonomous ordinary differential equation x = f (t, x) in the
situation when the vector field f is of limited regularity, typically belonging to
a space LP (O,T; Lq (JRn)). Such equations arise naturally when switching from an
Eulerian to a Lagrangian viewpoint for the solutions of partial differential equations.
We discuss some measurability issues in the foundations of the theory of regular
Lagrangian flow solutions. Further, we examine the sensitivity of the choice of
representative vector field f on solutions of the ordinary differential equation and,
in particular, we demonstrate that every vector field can be altered on a set of
measure zero to introduce non-uniqueness of solutions.
We develop some geometric tools to quantify the behaviour of solutions,
notably a non-autonomous version of subset avoidance and the r-codimension print
that encodes the dimension of a subset S c JRn x [0, T] while distinguishing between
the spatial and temporal detail of S. We relate this notion of dimension to the more
familiar box-counting dimensions, for which we prove some new inequalities.
Finally, motivated by the issues with measurability that can arise with irregular
vector fields we prove some fundamental results in the theory of Bochner
integration in order to be able to manipulate the representatives of the equivalence
classes in LP (O,T; Lq (JRn)).
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Differential equations, Vector fields | ||||
Official Date: | March 2012 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Robinson, James C. (James Cooper), 1969- | ||||
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Extent: | vi, 152 leaves : illustrations. | ||||
Language: | eng |
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