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Networks and Heterogeneous Media

2016, Volume 11, Issue 2: 203-222. doi: 10.3934/nhm.2016.11.203
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Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

  • 1.
    Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex
  • 2.
    Université Abdou Moumouni de Niamey, École Normale Supérieure, Departement de Mathématiques, BP: 10963 Niamey
  • Received: 01 May 2015 Revised: 01 October 2015

  • Primary: 35M13; Secondary: 35L04.

  • We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.

    Citation: Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws[J]. Networks and Heterogeneous Media, 2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203

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  • We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.


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