Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure

Benjamin Boutin; Frédéric Coquel; Philippe G. LeFloch; Benjamin Boutin; Frédéric Coquel; Philippe G. LeFloch

doi:10.3934/nhm.2021007

Networks and Heterogeneous Media

2021, Volume 16, Issue 2: 283-315. doi: 10.3934/nhm.2021007

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Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure

1.
Univ. Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
2.
Centre de Mathématique Appliquées, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
3.
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Sorbonne Université, 75258 Paris, France

Received: 01 October 2020 Revised: 01 January 2021 Published: 07 February 2021
Primary: 35L65, 35D40

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.
- Hyperbolic conservation law,
- coupling technique,
- Riemann problem,
- self-similar approximation,
- resonant effect
Citation: Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure[J]. Networks and Heterogeneous Media, 2021, 16(2): 283-315. doi: 10.3934/nhm.2021007

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Abstract

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.

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