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Approximation of entropy solutions to degenerate nonlinear parabolic equations

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Abstract

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space \(L^\infty \), whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and \(L^1\)-stability. We prove that the sequence of approximate solutions is strongly \(L^1\)-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.

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Authors and Affiliations

  1. Universidade Estadual de Campinas, Campinas, Brazil

    Eduardo Abreu

  2. Universidade de São Paulo, São Paulo, Brazil

    Mathilde Colombeau

  3. Universidade Estadual de Campinas, Campinas, Brazil

    Mathilde Colombeau

  4. Novgorod State University, Veliky Novgorod, Russian Federation

    Evgeny Yu Panov

  5. St. Petersburg State University, Saint Petersburg, Russian Federation

    Evgeny Yu Panov

Authors
  1. Eduardo Abreu
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  2. Mathilde Colombeau
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  3. Evgeny Yu Panov
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Correspondence to Mathilde Colombeau.

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Abreu, E., Colombeau, M. & Panov, E.Y. Approximation of entropy solutions to degenerate nonlinear parabolic equations. Z. Angew. Math. Phys. 68, 133 (2017). https://doi.org/10.1007/s00033-017-0877-6

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  • DOI: https://doi.org/10.1007/s00033-017-0877-6

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