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Generalized regularity and solution concepts for differential equations | | Simon Haller
; | | Date: |
9 Jun 2008 | | Abstract: | As the title ’’Generalized regularity and solution concepts for differential
equations’’ suggests, the main topic of my thesis is the investigation of
generalized solution concepts for differential equations, in particular first
order hyperbolic partial differential equations with real-valued, non-smooth
coefficients and their characteristic system of ordinary differential
equations.
In Colombeau theory there have been developed existence results that yield
solutions for ordinary and partial differential equations beyond the scope of
classical approaches. Nevertheless this comes at the price of sacrificing
regularity (in general a Colombeau solution may even lack a distributional
shadow). It is prevailing in the Colombeau setting that the question of mere
existence of solutions is much easier to answer than to determine their
regularity properties (i.e. if a distributional shadow exists and how regular
it is). In order order to address these regularity question and encouraged by
the fact that the solution of a (homogeneous) first order partial differential
equation can be written as a pullback of the initial condition by the
characteristic backward flow, a main topic of my thesis deals with the
microlocal analysis of pullbacks of c-bounded Colombeau generalized functions.
Another topic is the comparsion of Colombeau techniques for solving ordinary
and partial differential equations to other generalized solution concepts,
which has led to a joint article with G"unther H"ormann. A useful tool for
this purpose is the concept of a generalized graph, which has been developed in
the thesis. | | Source: | arXiv, 0806.1451 | | Services: | Forum | Review | PDF | Favorites | |
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