Communications in Mathematical Sciences

Volume 9 (2011)

Number 1

On the uniqueness of entropy solutions to the Riemann problem for 2×2 hyperbolic systems of conservation laws

Pages: 161 – 185

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n1.a8

Author

Hiroki Ohwa (Graduate School of Education, Waseda University, Tokyo, Japan)

Abstract

In this paper we revisit the Riemann problem for 2×2 hyperbolic systems of conservation laws, which satisfy the condition that the product of non-diagonal elements in the Fréchet derivative (Jacobian) of the flux is positive, the genuine nonlinearity condition, and the Smoller–Johnson condition in one space variable. The first condition implies that the system is strictly hyperbolic. By developing the shock curve approach, we give an alternative shock curve approach and re-prove the uniqueness of self-similar solutions satisfying the Lax entropy condition at discontinuities.

Keywords

conservation laws, the Riemann problem, shock approach

2010 Mathematics Subject Classification

35L65, 35L67, 58J45

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