Electron. J. Diff. Eqns., Vol. 2005(2005), No. 124, pp. 1-25.

Ill-posedness of the Cauchy problem for totally degenerate system of conservation laws

Wladimir Neves, Denis Serre

In this paper we answer some open questions concerning totally degenerate systems of conservation laws. We study the augmented Born-Infeld system, which is the Born-Infeld model augmented by two additional conservations laws. This system is a nice example of totally degenerate system of conservation laws and, global smooth solutions are conjectured to exist when the initial-data is smooth. We show that this conjecture is false, for the more natural and general condition of initial-data. In fact, first we show that does not exist global smooth solution for any 2X2 totally degenerated system of conservation laws, which the characteristics speeds do not have singular points. Moreover, we sharpen the conjecture in Majda [20]. Under the same hypothesis of initial-data, we show that the Riemann Problem is not well-posed, which follows for weak solutions of the Cauchy Problem. In the end, we prove some results on the direction of well-posedness for the less physically initial-data.

Submitted November 18, 2004. Published November 7, 2005.
Math Subject Classifications: 35L65, 46E30, 35L50, 26B20, 35L67, 26B12.
Key Words: Conservation laws; Cauchy problem; totally degenerated systems; ill-posed.

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Wladimir Neves
Instituto de Matematica, Universidade Federal do Rio de Janeiro
C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil
email: wladimir@im.ufrj.br
Denis Serre
UMPA, Ecole Normale Superieure de Lyon
UMR 5669 CNRS, Lyon Cedex 07, France
email: serre@umpa.ens-lyon.fr

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