Abstract
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.
Highlights
Many models in continuum physics involve augmentation terms containing small parameters such as the viscosity, heat conduction, relaxation effects, etc. These terms are typically modeled by second- or third-order derivatives which are taken into account in the fundamental conservation principles of continuum physics
On the other hand, when the third-order effects are dominant, highly oscillating patterns are observed near sharp gradients of the solutions
It is natural to assume that the orders of magnitude of the physical coefficients arising in a combination of terms like ν uxx +κ uxxx are such that the ratio ξ = κ/ν2 is of order 1
Summary
Many models in continuum physics involve augmentation terms containing small parameters such as the viscosity, heat conduction, relaxation effects, etc. These terms are typically modeled by second- or third-order derivatives which are taken into account in the fundamental conservation principles of continuum physics. For the present Note, the global dynamics of the shocks turn out to depend upon the small-scale physical modeling, and it is our aim here to provide a framework in which general classes of interest can be derived and analyzed. (which is not solution-dependent), together with a methodology in order to derive sufficient and necessary conditions ensuring that the augmentation terms are compatible with a given entropy. We directly introduce our key definition in the one-dimension context in Definition 1, below
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