Numerical analysis, Partial differential equations
The positive entropy production property for augmented nonlinear hyperbolic models
Comptes Rendus. Mathématique, Volume 360 (2022) no. G1, 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.

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Accepted:
Published online:
DOI: 10.5802/crmath.278
LeFloch, Philippe G. 1; Tesdall, Allen M. 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.
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LeFloch, Philippe G.; Tesdall, Allen M. The positive entropy production property for augmented nonlinear hyperbolic models. Comptes Rendus. Mathématique, Volume 360 (2022) no. G1, pp. 35-46. doi : 10.5802/crmath.278. http://www.numdam.org/articles/10.5802/crmath.278/

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