Boundary Value Problems
Volume 2011 (2011), Article ID 138396, 16 pages
Research Article

Two-Dimension Riemann Initial-Boundary Value Problem of Scalar Conservation Laws with Curved Boundary

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Educational Institutes, Jinan University, Guangzhou 510632, China

Received 16 December 2010; Accepted 1 February 2011

Academic Editor: Julio Rossi

Copyright © 2011 Huazhou Chen and Tao Pan. 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.


This paper is concerned with the structure of the weak entropy solutions to two-dimension Riemann initial-boundary value problem with curved boundary. Firstly, according to the definition of weak entropy solution in the sense of Bardos-Leroux-Nedelec (1979), the necessary and sufficient condition of the weak entropy solutions with piecewise smooth is given. The boundary entropy condition and its equivalent formula are proposed. Based on Riemann initial value problem, weak entropy solutions of Riemann initial-boundary value problem are constructed, the behaviors of solutions are clarified, and we focus on verifying that the solutions satisfy the boundary entropy condition. For different Riemann initial-boundary value data, there are a total of five different behaviors of weak entropy solutions. Finally, a worked-out specific example is given.