Conservation laws with coinciding smooth solutions but different conserved variables

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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.

Keywords

Hyperbolic conservation laws Compressible Euler equations Isentropic gas dynamics 

Mathematics Subject Classification

Primary 35L65 Secondary 35Q35 76N99 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.INDAM UnitUniversity of BresciaBresciaItaly
  2. 2.Department of Mathematics and Its ApplicationsUniversity of Milano-BicoccaMilanItaly

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