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On a Splitting-Differentiation Process Leading to Cross-Diffusion

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Progress in Industrial Mathematics at ECMI 2016 (ECMI 2016)

Part of the book series: Mathematics in Industry ((TECMI,volume 26))

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Abstract

We generalize the dynamical system model proposed by Sánchez-Palencia for the splitting-differentiation process of populations to include spatial dependence. This gives rise to a family of cross-diffusion partial differential equations problems, among which we consider the segregation model proposed by Busenberg and Travis. For the one-dimensional case, we make a direct parabolic regularization of the problem to show the existence of solutions in the space of BV functions. Moreover, we introduce a Finite Element discretization of both our parabolic regularization and an alternative regularization previously proposed in the literature. Our numerical results suggest that our approach is more stable in the tricky regions where the solutions exhibit discontinuities.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Universidad de Oviedo, c/Calvo Sotelo, 33007, Oviedo, Spain

    Gonzalo Galiano & Virginia Selgas

Authors
  1. Gonzalo Galiano
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  2. Virginia Selgas
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Corresponding author

Correspondence to Virginia Selgas .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Peregrina Quintela

  2. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Patricia Barral

  3. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Dolores Gómez

  4. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Francisco J. Pena

  5. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Jerónimo Rodríguez

  6. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Pilar Salgado

  7. Department of Applied Mathematics, University of Santiago de Compostela, Lugo, Spain, Spain

    Miguel E. Vázquez-Méndez

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Galiano, G., Selgas, V. (2017). On a Splitting-Differentiation Process Leading to Cross-Diffusion. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_62

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