Well--posedness for a class of hyperbolic systems of conservation laws in several space dimensions
Luigi Ambrosio, François Bouchut,and Camillo De Lellis
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Submission date: 25. Jul. 2003
published in: Communications in partial differential equations, 29 (2004) 9/10, p. 1635-1651
DOI number (of the published article): 10.1081/PDE-200040210
MSC-Numbers: 35L45, 35L40, 35L65
Keywords and phrases: hyperbolic systems, several dimensions, renormalized solutions
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In this paper we consider a system of conservation laws in several space dimensions whose nonlinearity is due only to the modulus of the solution. This system, first considered by Keyfitz and Kranzer in one space dimension, has been recently studied by many authors. In particular, using standard methods from DiPerna--Lions theory, we improve the results obtained by the first and third author, showing existence, uniqueness and stability results in the class of functions whose modulus satisfies, in the entropy sense, a suitable scalar conservation law. In the last part of the paper we consider a conjecture on renormalizable solutions and show that this conjecture implies another one recently made by Bressan in connection with the system of Keyfitz and Kranzer.