# American Institute of Mathematical Sciences

June  2021, 16(2): 283-315. doi: 10.3934/nhm.2021007

## 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

Received  October 2020 Revised  January 2021 Published  February 2021

Fund Project: 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)

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.

Citation: Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
##### References:
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##### References:
 [1] Ad imurthi, S. Mishra and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.  Google Scholar [2] Ad imurthi, S. Mishra and G. D. V. Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2 (2007), 127-157.  doi: 10.3934/nhm.2007.2.127.  Google Scholar [3] Ad imurthi, S. Mishra and G. D. V. Gowda, Conservation law with the flux function discontinuous in the space variable. Ⅱ. Convex-concave type fluxes and generalized entropy solutions, J. Comput. Appl. Math., 203 (2007), 310-344.  doi: 10.1016/j.cam.2006.04.009.  Google Scholar [4] A. Ambroso, C. Chalons, F. Coquel and T. Galié, Interface model coupling via prescribed local flux balance, ESAIM Math. Model. Numer. Anal., 48 (2014), 895-918.  doi: 10.1051/m2an/2013125.  Google Scholar [5] B. 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Cancès, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12 (2015), 343-384.  doi: 10.1142/S0219891615500101.  Google Scholar [9] B. Andreianov, K. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.  doi: 10.3934/nhm.2010.5.617.  Google Scholar [10] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.  Google Scholar [11] B. Andreianov and D. Mitrović, Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1307–1335. doi: 10.1016/j.anihpc.2014.08.002.  Google Scholar [12] B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964.  doi: 10.3934/dcds.2012.32.1939.  Google Scholar [13] E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.  doi: 10.1017/S0308210500003863.  Google Scholar [14] C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.  Google Scholar [15] M. Benyahia, C. Donadello, N. Dymski and M. D. Rosini, An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 48-90.  doi: 10.1007/s00030-018-0539-1.  Google Scholar [16] C. Berthon, M. Bessemoulin-Chatard, A. Crestetto and F. Foucher, A Riemann solution approximation based on the zero diffusion-dispersion limit of Dafermos reformulation type problem, Calcolo, 56 (2019), 28-60.  doi: 10.1007/s10092-019-0325-4.  Google Scholar [17] C. Berthon, F. Coquel and P. G. LeFloch, Why many theories of shock waves are necessary: Kinetic relations for non-conservative systems, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1-37.  doi: 10.1017/S0308210510001009.  Google Scholar [18] B. Boutin, C. Chalons and P.-A. Raviart, Existence result for the coupling problem of two scalar conservation laws with Riemann initial data, Math. Models Methods Appl. Sci., 20 (2010), 1859-1898.  doi: 10.1142/S0218202510004817.  Google Scholar [19] B. Boutin, F. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅰ: Self-similar diffusion for thin interfaces, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 921-956.  doi: 10.1017/S0308210510001459.  Google Scholar [20] B. Boutin, F. 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Real World Appl., 40 (2018), 403-427.  doi: 10.1016/j.nonrwa.2017.09.005.  Google Scholar [29] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.  doi: 10.1016/0022-247X(72)90114-X.  Google Scholar [30] C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal., 52 (1973), 1-9.  doi: 10.1007/BF00249087.  Google Scholar [31] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, fourth edition, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar [32] G. Dal Maso, P. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), 483-548.   Google Scholar [33] S. 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Illustration of case (ⅱ) of the Corollary 1. The solution $u$ is continuous at $\xi = 0$ (left) vs. discontinuous at $\xi = 0$ (right)
Standing shocks and non trivial inner solution
Non matching property
Structure of double-rarefaction solutions
Structure of double-shock solutions
Candidate CRD solutions for two convex quadratic fluxes (case $c>0$)
Candidate CRD solutions for two convex quadratic fluxes (case $c<0$)
Numerical approximation of the double-shock solution. CRD solution $u$ (left) and selection criterion $h(\cdot;u)$ (right)
Numerical approximation of the double-rarefaction solution. CRD solution $u$ (left) and selection criterion $h(\cdot;u)$ (right)
Three admissible solutions (from top to bottom). CRD solution $u$ (left) and selection criterion $h(\cdot;u)$ (right)
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