Abstract
Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm–Lax result (Glimm and Lax in Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, No. 101. American Mathematical Society, Providence, 1970), we obtain estimates improving those in Saint-Raymond (Arch Ration Mech Anal 155(3):171–199, 2000) on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model.
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The present work was supported by the PRIN 2015 Project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications, by the GNAMPA 2017 Project Conservation Laws: from Theory to Technology and by the Simons Foundation Grant 346300 together with the Polish Government MNiSW 2015–2019 matching fund.
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Colombo, R.M., Guerra, G. Conservation laws with coinciding smooth solutions but different conserved variables. Z. Angew. Math. Phys. 69, 47 (2018). https://doi.org/10.1007/s00033-018-0942-9
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DOI: https://doi.org/10.1007/s00033-018-0942-9