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2001 Weak Solutions for a Simple Hyperbolic System
Owen Lyne, David Williams
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Electron. J. Probab. 6: 1-21 (2001). DOI: 10.1214/EJP.v6-93


The model studied concerns a simple first-order hyperbolic system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as weak solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical `exact' solution which is everywhere defined. The direct method used is guided by the theory of measure-valued diffusions. The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to Itô's formula.

We then conduct computer studies of our model, both by integration schemes (which do use characteristics) and by `random simulation'.


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Owen Lyne. David Williams. "Weak Solutions for a Simple Hyperbolic System." Electron. J. Probab. 6 1 - 21, 2001.


Accepted: 15 August 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 0984.35101
MathSciNet: MR1873297
Digital Object Identifier: 10.1214/EJP.v6-93

Primary: 35L35
Secondary: 60G44 , 60J27

Keywords: Branching processses , Martingales , Travelling waves , weak solutions


Vol.6 • 2001
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