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Einladung zur Defensio von Christian Spreitzer

Freitag, 30. Juni 2017, 10:15Uhr

Seminarraum 11, 2. Stock, Oskar-Morgenstern-Platz 1, 1090 Wien

Einladung

Abstract

Hyperbolic systems of partial differential equations: Generalised solutions, regularity, and regularisations

In this thesis, we present existence and uniqueness results for the Cauchy problem for symmetric hyperbolic systems of PDEs in Colombeau algebras of generalised functions. This framework allows us to consider equations with highly singular coefficients, for which distributional solution concepts fail. The essential methods are based on energy estimates on lens-shaped domains in space-time. We obtain distributional limits of generalised solutions even in cases where the coefficient regularities are below those required within classical weak solution concepts. As an application, we consider the Dirac equation with a delta distribution coefficient. We establish the transformation from a scalar second-order wave equation to a related first-order system in a generalised functions setting, thereby obtaining existence and uniqueness results also for second-order wave equations. These results can be applied to wave equations derived from the Laplace-Beltrami operator of metrics of low regularity. Finally, we develop an explicit construction of Schwartz functions with prescribed moments and support contained in a conic set, providing a constructive solution to a variant of the Stieltjes moment problem as well as tailor-made mollifiers for regularisations of distributions.  

Prüfungssenat:

Michael Oberguggenberger, Universität Innsbruck, A (Gutachter)

James A. Vickers, University of Southampton, UK (Gutachter)

Günther Hörmann (Betreuer)

Martin Hopf (Vorsitz)

SSC Physik
Universität Wien
Boltzmanngasse 5
1090 Wien

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