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The Scientific World Journal
Volume 2014, Article ID 580871, 5 pages
http://dx.doi.org/10.1155/2014/580871
Research Article

Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

Received 16 January 2014; Accepted 2 March 2014; Published 30 March 2014

Academic Editors: D. Baleanu and C. M. Khalique

Copyright © 2014 Sen Wong and Manwai Yuen. 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 study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in . For time , we can define a functional associated with the solution of the equations and some testing function . When the pressure function of the governing equations is of the form , where is the density function, is a constant, and , we can show that the nontrivial solutions with nonslip boundary condition will blow up in finite time if satisfies some initial functional conditions defined by the integrals of . Examples of the testing functions include , , , , and . The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.