Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 124, pp. 1-25.
Title: Ill-posedness of the Cauchy problem for totally degenerate
system of conservation laws
Authors: Wladimir Neves (Univ. Federal do Rio de Janeiro, Brazil)
Denis Serre (Ecole Normale Superieure de Lyon, France)
Abstract:
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.