Abstract: 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 condition and prove the existence of global smooth solutions for the systems with initial data close to constant equilibrium states where the entropy condition and Kawashima's condition are satisfied. In addition, we show that a system of balance laws satisfies Kawashima's condition if and only if so does its first-order approximation---a hyperbolic-parabolic system---derived from the Chapman-Enskog expansion. The result is applied to Bouchut's discrete velocity BGK models approximating hyperbolic systems of conservation laws.
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