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Euler-Lagrange change of variables in conservation laws. (English) Zbl 1132.35060

The author introduces a method for studying the Cauchy problem for systems of conservation laws in one space dimension. The method is based on the equivalence of the problems in the Euler and the Lagrange coordinates. It is proved that the Cauchy problem in these coordinates are equivalent as regards to existence and uniqueness of entropy solutions in \(L^\infty\). An explicit relation between solutions is exhibited. Some applications of the suggested approach are considered: the Keyfitz-Kranzer system, linear Lagrange systems (including the system of pressureless gas dynamics, \(2\times 2\) linear degenerate systems, and the augmented Born-Infeld equations), and the Born-Infeld system (not linear in Lagrange coordinates).

MSC:

35L65 Hyperbolic conservation laws
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
76N15 Gas dynamics (general theory)
35Q35 PDEs in connection with fluid mechanics
35L45 Initial value problems for first-order hyperbolic systems
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