Entropy and global existence for hyperbolic balance laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchut's discrete velocity BGK models approximating hyperbolic systems of conservation laws.