International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 101647, 16 pages
doi:10.1155/2009/101647
Research Article

Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

Antonio Olivas Martinez and Georgy A. Omel'yanov

Departamento de Matematicas, Universidad de Sonora, Calle Rosales y Blvd Luis Encinas, s/n, 83000, Hermosillo, Sonora, Mexico

Received 7 July 2009; Revised 11 November 2009; Accepted 9 December 2009

Academic Editor: Ingo Witt

Copyright © 2009 Antonio Olivas Martinez and Georgy A. Omel'yanov. 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

We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.