Relative Entropy for Hyperbolic–Parabolic Systems and Application to the Constitutive Theory of Thermoviscoelasticity

Abstract

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L p framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

Dissipative measure-valued solutions for general conservation laws

“…Thus one can provide a simple derivation of weak limit as a Young measure and a concentration measure. Under slightly different assumptions on the entropy and in the same formulation as currently considered, i.e., A(u) is not necessarily an identity, as in the aforementioned results, the issue of measurevalued-strong uniqueness was considered in [7]. In the spirit of these results, the issue of mv-strong uniqueness was considered for various systems, including compressible Euler system and Savage-Hutter system describing granular media in [21], compressible Navier-Stokes in [17] and complete compressible Euler system in [5].…”

“…The relative entropy method, which is fundamental for mv-strong uniqueness results, appears to be useful for other areas such as stability studies, asymptotic limits and dimension reduction problems (e.g. [7], [20], [18], [3], [6]). Not only the systems describing phenomena of mathematical physics fall into these applications.…”

A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

“…The class of symmetrizable systems encompasses important examples from applications -most notably the equations of gas dynamics -and were singled out as a class by Lax and Friedrichs [17]. The remarkable stability properties induced by convex entropies are captured by the relative entropy method of Dafermos [8,7] and DiPerna [13], as extended for the system (1.1)-(1.2) by Christoforou-Tzavaras [5]. However, many thermomechanical systems do not fit under the framework of symmetrizable systems.…”

“…and we discuss the stability of systems in thermomechanics when they are generated by polyconvex free energies. Following [20,11], we extend the system into an augmented symmetrizable system (see Section 2), and then apply to the resulting system the relative entropy formulation following [5]. This allows to carry out the convergence results from thermoviscoelasticity to adiabatic thermoelasticity, or from thermoviscoelasticity to thermoelasticity under the framework of hypetheses (1.9)-(1.10).…”

Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

“…The relative entropy method of Dafermos [11,10] and DiPerna [15] provides an analytical framework upon which one can examine such questions, and has been tested in a variety of contexts (e.g. [5,14,20,8,18,6]).…”