Solution of conservation laws via convergence space completion

It is well known that a classical solution of the initial value problem for a scalar conservation law may fail to exist on the whole domain of definition of the problem. For this reason, suitable generalized solutions of such problems, known as weak solutions, have been considered and studied extensively. However, weak solutions are not unique. In order to obtain a unique solution that is physically relevant, the vanishing viscosity method, amongst others, has been employed to single out a unique solution known as the entropy solution. In this thesis we present an alternative approach to the study of the entropy solution of conservation laws. The main novelty of our approach is that the theory of entropy solution of conservation law is presented in an operator theoretic setting. In this regard, the Order Completion Method for nonlinear PDEs, in the context of convergence vector spaces, is modified to obtain an operator equation which generalizes the initial value problem. This equation admits at most one solution, which may be represented as a Hausdorff continuous function. As a particular case, we apply our method to obtain the entropy solution of the Burger's equation. Copyright

Grantor dc:publisher

University of Pretoria

Year dc:date.issued

2011

Advisors dc:contributor.advisor

• Van der Walt, Jan Harm
• Anguelov, Roumen

• Conservation laws
• Entropy solution
• Order completion
• Convergence space
• Wyler completion
• Hausdorff continuous function.
• UCTD

Statement dc:rights

• © 2011, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria

Dc Identifier Other

C12/4/110/gm

OAI identifier oai:identifier

oai:repository.up.ac.za:2263/27791