Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Blowup in $\mathbf{L^{\infty}}$ for a class of genuinely nonlinear hyperbolic systems of conservation laws

Pages: 837 - 853, Volume 7, Issue 4, October 2001      doi:10.3934/dcds.2001.7.837

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Paolo Baiti - Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Udine 33100, Italy (email)
Helge Kristian Jenssen - Mathematics Department, Indiana University, Rawles Hall, Bloomington, IN 47405, United States (email)

Abstract: We construct a class of $3\times 3$ systems of conservation laws with all characteristic fields genuinely nonlinear, and we show the existence of entropy solutions for these that blow up in sup-norm in finite time. The solutions are constructed by considering wave patterns where infinitely many shock waves are produced in finite time. We also consider the role of entropies as a mechanism for preventing this type of singular behavior.

Keywords:  Systems of conservation laws, hyperbolic, genuine nonlinear, blowup.
Mathematics Subject Classification:  35L65, 35B05.

Received: September 2000;      Revised: February 2001;      Available Online: July 2001.