Electron. J. Diff. Eqns.,
Vol. 2005(2005), No. 124, pp. 125.
Illposedness of the Cauchy problem for totally degenerate
system of conservation laws
Wladimir Neves, Denis Serre
Abstract:
In this paper we answer some open questions concerning totally
degenerate systems of conservation laws. We study the augmented
BornInfeld system, which is the BornInfeld 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 initialdata is
smooth. We show that this conjecture is false, for the more
natural and general condition of initialdata. 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 initialdata, we show that the Riemann Problem is
not wellposed, which follows for weak solutions of the Cauchy
Problem. In the end, we prove some results on the direction of
wellposedness for the less physically initialdata.
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; illposed.
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Wladimir Neves
Instituto de Matematica, Universidade Federal do Rio de
Janeiro
C. Postal 68530, Rio de Janeiro, RJ 21945970, Brazil
email: wladimir@im.ufrj.br


Denis Serre
UMPA, Ecole Normale Superieure de Lyon
UMR 5669 CNRS, Lyon Cedex 07, France
email: serre@umpa.enslyon.fr

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