As already noted in Section 7.9, the function space of choice for weak solutions of hyperbolic systems of conservation laws in one space dimension is BV, since it is within its confines that one may discern shocks and study their propagation and interactions. The notion of characteristic, introduced in Section 7.2 for classical solutions, will here be extended to the framework of BV weak solutions. It will be established that generalized characteristics propagate either with classical characteristic speed or with shock speed. In particular, it will be shown that the extremal backward characteristics, emanating from any point in the domain of an admissible solution, always propagate with classical characteristic speed. The implications of these properties to the theory of weak solutions will be demonstrated in following chapters.
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