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

Benjamin Boutin; Frédéric Coquel; Philippe G. LeFloch

doi:10.3934/nhm.2021007

Article Contents

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

^* Corresponding author: Benjamin Boutin
^* Corresponding author: Benjamin Boutin

Received: September 30, 2020

Revised: December 31, 2020

Early access: February 2021

Published: June 2021

The three authors were partially supported by the Innovative Training Networks (ITN) grant 642768 (ModCompShock), and by the Centre National de la Recherche Scientifique (CNRS)

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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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Mathematics Subject Classification: Primary: 35L65, 35D40.

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Figure 1. Illustration of case (ⅱ) of the Corollary 1. The solution $ u $ is continuous at $ \xi = 0 $ (left) vs. discontinuous at $ \xi = 0 $ (right)

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Figure 2. Standing shocks and non trivial inner solution

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Figure 3. Non matching property

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Figure 4. Structure of double-rarefaction solutions

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Figure 5. Structure of double-shock solutions

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Figure 6. Candidate CRD solutions for two convex quadratic fluxes (case $ c>0 $)

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Figure 7. Candidate CRD solutions for two convex quadratic fluxes (case $ c<0 $)

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Figure 8. Numerical approximation of the double-shock solution. CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)

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Figure 9. Numerical approximation of the double-rarefaction solution. CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)

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Figure 10. Three admissible solutions (from top to bottom). CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)

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