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Open Access Research Article

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

Huazhou Chen12 and Tao Pan2*

Author Affiliations

1 Department of Mathematics, Shanghai University, Shanghai 200444, China

2 Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Educational Institutes, Jinan University, Guangzhou 510632, China

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Boundary Value Problems 2011, 2011:138396  doi:10.1155/2011/138396

Published: 24 February 2011

Abstract

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.