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General hyperbolic systems of balance laws with inhomogeneity in space andtime in all constitutive functions are studied in the context of relativeentropy. A framework is developed in this setting that contributes to ameasure-valued weak vs strong uniqueness theorem, a stability theorem ofviscous solutions and a convergence theorem as the viscosity parameter tends tozero. The main goal of this paper is to develop hypotheses under which therelative entropy framework can still be applied. Examples of systems withinhomogeneity that have different charateristics are presented and thehypotheses are discussed in the setting of each example.

The Relative Entropy Method for Inhomogeneous Systems of Balance Laws