Topological resolution of Riemann problems for pairs of conservation laws
Authors:
Arthur V. Azevedo, Cesar S. Eschenazi, Dan Marchesin and Carlos F. B. Palmeira
Journal:
Quart. Appl. Math. 68 (2010), 375393
MSC (2000):
Primary 35L65, 58J45, 35L60, 57R99
Published electronically:
February 19, 2010
MathSciNet review:
2663005
Fulltext PDF
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Abstract: The structure of Riemann solutions to certain systems of conservation laws can be so complicated that the classical constructions are unable to establish their global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point represents the intermediate constant state that typically appears in such solutions. In this paper, we construct the wave curves in a threedimensional manifold which is globally foliated by shock curves, and where rarefaction and composite curves are naturally defined. The main innovation in this paper is the construction of a differentiable twodimensional manifold of intermediate states; the local classical construction is replaced by finding the intersection of a wave curve with the intermediate manifold; a transversality argument guarantees the stability of the Riemann solution. The new construction has the potential of establishing structural stability properties globally, i.e., for all initial data. This is its main advantage over the classical construction, which is intrinsically local. The construction presented in this work is demonstrated for a family of quadratic polynomial flux functions and their perturbations.
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Additional Information
Arthur V. Azevedo
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910900 Brasília, DF, Brazil
Email:
arthur@mat.unb.br
Cesar S. Eschenazi
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, Avenida Antônio Carlos 6627, 31270901 Belo Horizonte, MG, Brazil
Email:
cesar@mat.ufmg.br
Dan Marchesin
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada D. Castorina 110, 22460320 Rio de Janeiro, RJ, Brazil
Email:
marchesi@impa.br
Carlos F. B. Palmeira
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, 22453900 Rio de Janeiro, RJ, Brazil
Email:
fredpalm@mat.pucrio.br
DOI:
http://dx.doi.org/10.1090/S0033569X10011547
PII:
S 0033569X(10)011547
Keywords:
Conservation laws,
Riemann problem,
wave curves,
wave manifold,
intermediate surface
Received by editor(s):
October 23, 2008
Published electronically:
February 19, 2010
Additional Notes:
The first author was supported in part by FEMAT under Grant 04/10, UnB under Grant FUNPE 2005, and CNPq/PADCT under Grant 620029/20048
The second author was supported in part by CNPq under Grant 151363/20072 and FAPERJ under Grant E26/150533/2007
The third author was supported in part by CNPq under Grants 472067/20060, 304168/20068 and 491148/20054, and by FAPERJ under Grants E26/152.525/2006, E26/110.310/2007.
Article copyright:
© Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.
