Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Richness or semi-Hamiltonicity of quasi-linear systems that are not in evolution form


Author: Misha Bialy
Journal: Quart. Appl. Math. 71 (2013), 787-796
MSC (2000): Primary 35L65, 35L67, 70H06
DOI: https://doi.org/10.1090/S0033-569X-2013-01327-3
Published electronically: August 30, 2013
MathSciNet review: 3136996
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Abstract: The aim of this paper is to consider strictly hyperbolic quasi-linear systems of conservation laws which appear in the form $ A(u)u_{x}+B(u)u_{y}=0.$ If one of the matrices $ A(u), B(u)$ is invertible, then this system is in fact in the form of evolution equations. However, it may happen that traveling along characteristics one moves from the ``chart'' where $ A(u)$ is invertible to another ``chart'' where $ B(u)$ is invertible. We propose a new condition of richness or semi-Hamiltonicity for such a system that is ``chart''-independent. This new condition enables one to perform the blow-up analysis along characteristic curves for all times, not passing from one ``chart'' to another. This opens a possibility to use this theory for geometric problems as well as for stationary solutions of 2D+1 systems. We apply the results to the problem of polynomial integral for geodesic flows on the 2-torus.


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Additional Information

Misha Bialy
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
Email: bialy@post.tau.ac.il

DOI: https://doi.org/10.1090/S0033-569X-2013-01327-3
Keywords: Rich, conservation laws, genuine nonlinearity, blow-up, systems of hydrodynamic type
Received by editor(s): March 26, 2012
Published electronically: August 30, 2013
Additional Notes: Partially supported by ISF grant 128/10
Article copyright: © Copyright 2013 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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