SBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function

Abstract: We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws having small bounded variation, in one space dimension, must be functions of special bounded variation. For more than one equation, this is new also in the case of systems of conservation laws out of the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of balance laws: we generalize the SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measures, localization arguments, and observations in real analysis.