Comptes Rendus
Numerical analysis, Partial differential equations
The positive entropy production property for augmented nonlinear hyperbolic models
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 35-46.

Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, after introducing a notion of positive entropy production property that involves test-functions (rather than solutions), we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model viscosity and other high-order effects in a fluid. By introducing a new notion of positive entropy production that concerns general functions rather than solutions, we can easily check the entropy-dissipating property for a broad class of augmented models. The weak solutions associated with the corresponding zero-limit may contain (nonclassical undercompressive) shocks whose selection is determined from these high-order effects, for instance by using traveling wave solutions. Having a classification of the underlying models, as we propose, is essential for developing a general shock wave theory.

Published online:
DOI: 10.5802/crmath.278

Philippe G. LeFloch 1; Allen M. Tesdall 2

1 Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique Sorbonne Université, 4 Place Jussieu, 75252 Paris, France.
2 Department of Mathematics, City University of New York, College of Staten Island, and Physics Program, The Graduate Center, City University of New York, New York, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Philippe G. LeFloch and Allen M. Tesdall},
     title = {The positive entropy production property for augmented nonlinear hyperbolic models},
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Philippe G. LeFloch; Allen M. Tesdall. The positive entropy production property for augmented nonlinear hyperbolic models. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 35-46. doi : 10.5802/crmath.278.

[1] Rohan Abeyaratne; James K. Knowles Kinetic relations and the propagation of phase boundaries in solids, Arch. Ration. Mech. Anal., Volume 114 (1991) no. 2, pp. 119-154 | DOI | MR | Zbl

[2] Paolo Antonelli; Pierangelo Marcati On the finite energy weak solutions to a system in quantum fluid dynamics, Commun. Math. Phys., Volume 287 (2009) no. 2, pp. 657-686 | DOI | MR | Zbl

[3] Constantine M. Dafermos Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer, 2000 | DOI | Zbl

[4] Haitao Fan; Marshall Slemrod The Riemann problem for systems of conservation laws of mixed type, Shock induced transitions and phase structures in general media. Papers of a workshop, held in Minneapolis, MN, USA, Oct. 1990 (J. E. Dunn, ed.) (The IMA Volumes in Mathematics and its Applications), Volume 52, Springer, 1993, pp. 61-91 | MR | Zbl

[5] Henri Gouin; Sergey Gavrilyuk Dissipative two-fluid models, Suppl. Rend. Circ. Mat. Palermo, Volume 78 (2006), pp. 133-145 | Zbl

[6] Brian T. Hayes; Philippe G. LeFloch Nonclassical shocks and kinetic relations. Scalar conservation laws, Arch. Ration. Mech. Anal., Volume 139 (1997) no. 1, pp. 1-56 | DOI | Zbl

[7] Peter D. Lax Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conference Series in Applied Mathematics, 11, Society for Industrial and Applied Mathematics, 1973 | Zbl

[8] Philippe G. LeFloch Propagating phase boundaries: Formulation of the problem and existence via the Glimm scheme, Arch. Ration. Mech. Anal., Volume 123 (1993) no. 2, pp. 153-197 | DOI | Zbl

[9] Philippe G. LeFloch An introduction to nonclassical shocks of systems of conservation laws, An introduction to recent developments in theory and numerics for conservation laws. Proceedings of the international school, Freiburg/ Littenweiler, Germany, October 20–24, 1997 (Lecture Notes in Computational Science and Engineering), Volume 5, Springer, 1999, pp. 28-72 | MR | Zbl

[10] Philippe G. LeFloch Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, Lecture Notes in Mathematics, Birkhäuser, 2002 | Zbl

[11] Philippe G. LeFloch Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues, Nonlinear partial differential equations and hyperbolic wave phenomena. The 2008–2009 research program on nonlinear partial differential equations, Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway (Contemporary Mathematics), Volume 526, American Mathematical Society, 2010, pp. 237-272 | MR | Zbl

[12] Philippe G. LeFloch; Majid Mohammadian Why many theories of shock waves are necessary: Kinetic functions, equivalent equations, and fourth-order models, J. Comput. Phys., Volume 227 (2008) no. 8, pp. 4162-4189 | DOI | MR | Zbl

[13] Philippe G. LeFloch; Allen M. Tesdall Augmented hyperbolic models and diffusive-dispersive shocks (in preparation)

[14] Marshall Slemrod Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal., Volume 81 (1983), pp. 301-315 | DOI | MR | Zbl

[15] Lev Truskinovsky Dynamics of non-equilibrium phase boundaries in a heat conducting non- linear elastic medium, J. Appl. Math. Mech., Volume 51 (1987) no. 6, pp. 777-784 | DOI | Zbl

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