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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 809095, 9 pages
http://dx.doi.org/10.1155/2014/809095
Research Article

New Distributional Global Solutions for the Hunter-Saxton Equation

CMAF, University of Lisbon, Avenida Professor Gama Pinto 2, 1649-003 Lisbon, Portugal

Received 7 July 2014; Revised 7 September 2014; Accepted 8 September 2014; Published 14 October 2014

Academic Editor: Ziemowit Popowicz

Copyright © 2014 C. O. R. Sarrico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation in a convenient space of discontinuous functions. With the help of a consistent extension of the classical solution concept, two classes of discontinuous solutions are obtained: one class of conservative solutions and another of dispersive solutions. A necessary and sufficient condition for the propagation of a distributional profile as a travelling wave is also presented, which allows identifying an interesting set of explicit distributional travelling waves. In the paper, we will show some results we have obtained by applying this framework to other equations and systems.