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), 375-393

MSC (2000):
Primary 35L65, 58J45, 35L60, 57R99

Published electronically:
February 19, 2010

MathSciNet review:
2663005

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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 three-dimensional 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 two-dimensional 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, 70910-900 Brasília, DF, Brazil

Email:
arthur@mat.unb.br

**Cesar S. Eschenazi**

Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, Avenida Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazil

Email:
cesar@mat.ufmg.br

**Dan Marchesin**

Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada D. Castorina 110, 22460-320 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, 22453-900 Rio de Janeiro, RJ, Brazil

Email:
fredpalm@mat.puc-rio.br

DOI:
https://doi.org/10.1090/S0033-569X-10-01154-7

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 620029/2004-8

The second author was supported in part by CNPq under Grant 151363/2007-2 and FAPERJ under Grant E-26/150533/2007

The third author was supported in part by CNPq under Grants 472067/2006-0, 304168/2006-8 and 491148/2005-4, and by FAPERJ under Grants E-26/152.525/2006, E-26/110.310/2007.

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Brown University

The copyright for this article reverts to public domain 28 years after publication.