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.
Revised:
Accepted:
Published online:
Philippe G. LeFloch 1; Allen M. Tesdall 2
@article{CRMATH_2022__360_G1_35_0, author = {Philippe G. LeFloch and Allen M. Tesdall}, title = {The positive entropy production property for augmented nonlinear hyperbolic models}, journal = {Comptes Rendus. Math\'ematique}, pages = {35--46}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.278}, language = {en}, }
TY - JOUR AU - Philippe G. LeFloch AU - Allen M. Tesdall TI - The positive entropy production property for augmented nonlinear hyperbolic models JO - Comptes Rendus. Mathématique PY - 2022 SP - 35 EP - 46 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.278 LA - en ID - CRMATH_2022__360_G1_35_0 ER -
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.278/
[1] 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] 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] Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer, 2000 | DOI | Zbl
[4] 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] Dissipative two-fluid models, Suppl. Rend. Circ. Mat. Palermo, Volume 78 (2006), pp. 133-145 | Zbl
[6] Nonclassical shocks and kinetic relations. Scalar conservation laws, Arch. Ration. Mech. Anal., Volume 139 (1997) no. 1, pp. 1-56 | DOI | Zbl
[7] 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] 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] 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] Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, Lecture Notes in Mathematics, Birkhäuser, 2002 | Zbl
[11] 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] 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] Augmented hyperbolic models and diffusive-dispersive shocks (in preparation)
[14] 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] 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
Cited by Sources:
Comments - Policy