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Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems

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Abstract

We study the bifurcation of Riemann solutions due to parameter change that alters the type of an umbilic point existing in state space. Solutions with data near generic umbilic points are primarily determined by the local quadratic expansion of flux functions. We observe that near an umbilic point, the bifurcation of the solution is essentially local and its behavior depends solely on the cubic expansion of the flux functions. These phenomena are illustrated for immiscible three-phase flow in porous media, which looses strict hyperbolicity at an isolated point in the interior of the oil–water–gas saturation triangle.

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Authors and Affiliations

  1. Faculdade de Economia, Centro de Matemática Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    V. Matos

  2. Departamento de Matemática, Universidade de Brasília, 70910, Brasília, DF, Brazil

    A. V. Azevedo

  3. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, Brazil

    J. C. Da Mota

  4. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

    D. Marchesin

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  1. V. Matos
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  2. A. V. Azevedo
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  3. J. C. Da Mota
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  4. D. Marchesin
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Correspondence to V. Matos.

Additional information

This work was partially supported by CNPq under Grants PCI 302791/2011-6, PCI 170181/2013-8; 479186/06-5; 301564/2009-4, 470635/2012-6, 490707/2008-4, 402299/2012-4; FCT under Grant SFRH/BSAB/1164/2011; FAPEG under grant 005/2012-Universal FAPERJ under Grants E-26/111.416/2010, E-26/102.965/2011, E-26/110.658/2012, E-26/111.369/2012, E-26/1110.114/2013; ANP-PRH32-731948/2010; Petrobras-PRH32-6000.0069459.11.4. The first three authors gratefully acknowledge the hospitality of IMPA during this work.

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Matos, V., Azevedo, A.V., Da Mota, J.C. et al. Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems. Z. Angew. Math. Phys. 66, 1413–1452 (2015). https://doi.org/10.1007/s00033-014-0469-7

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  • DOI: https://doi.org/10.1007/s00033-014-0469-7

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