A Convex Entropy for a Hyperbolic System with Relaxation

We explicitly construct a convex entropy function for a hyperbolic system with relaxation. This entropy is defined in the sense of Chen, Levermore, and Liu, which resembles Boltzmanʹs H-Theorem for kinetic equations. This construction follows the idea of Suliciuʹs energy function for a rate-type mixed hyperbolic-elliptic system. Such an entropy will be useful in proving the entropy property of the relaxation schemes introduced by Jin and Xin for conservation laws.