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

Paolo Baiti; Helge Kristian Jenssen

doi:10.3934/dcds.2001.7.837

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2001, Volume 7, Issue 4: 837-853. Doi: 10.3934/dcds.2001.7.837

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

Paolo Baiti^1, and
Helge Kristian Jenssen^2,

1.
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Udine 33100
2.
Mathematics Department, Indiana University, Rawles Hall, Bloomington, IN 47405

Received: September 2000

Revised: February 2001

Published: October 2001

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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.

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Mathematics Subject Classification: 35L65, 35B05.

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