Relative entropy in hyperbolic relaxation for balance laws

Abstract

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [23] for systems of hyperbolic conservation laws. IntroductionWe present a general framework for the approximation of systems of hyperbolic balance laws,by relaxation systems presented in the form of the extended systemin the regime where the solution of the limiting system (as ε → 0) is smooth. Motivated by the structure of physical models and the analysis in [6], we deal with relaxation systems of type (1.2) which are equipped with a globally defined, convex entropy H(U ) satisfyingThe problem of numerical approximation of nonlinear hyperbolic balance laws is extremely challenging. In the present article we identify a class of relaxation schemes suitable for the approximation of solutions to certain systems of hyperbolic balance laws arising in continuum physics. The

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

“…Hypothesis (H d ) has been proposed in a context of relaxation balance laws in [22] and can be easily verified for the example of friction in gas dynamics studied in [21]. Hypothesis (A G ) will lead via an argument as in (A.3) to the bound G(u|ū) ≤ Cη(u|ū).…”

“…[11,3,10,26]), or balance laws (e.g. [29,22]), and has recently being applied to problems that are classified under the domain of hyperbolicparabolic systems (e.g [13,20,21,4]). The objective of this work is to systematize the derivation of relative entropy identities, referring to (1.1) as a unifying framework, in order to connect the relative entropy theory with its natural framework, the L 2 theory of hyperbolic-parabolic systems of Kawashima [18] and the developments on Green functions by Liu-Zeng [30], and, even further, to exhibit the intimate connection of this method with the framework of thermodynamics.…”

On The Relative Entropy Method For Hyperbolic-Parabolic Systems

“…[12,3,11,24]), or balance laws (e.g. [27,23]), and has recently being applied to problems that are classified under the domain of hyperbolic-parabolic systems (e.g [14,21,22,4]). The aim is to extend the class of computations that go under the general term relative entropy to the broader class of systems (2) and (1) and to systematize the derivation of relative entropy identities in a unifying framework.…”

Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

“…In this section we address the problem of relaxation to the isentropic combustion model We impose the following requirements on P, Θ (see [16] for the motivation):…”

“…The reader should contrast the relaxation system presented in this work with other relaxation systems proposed in the literature [16,17], where relaxation approximations to hyperbolic balance laws were proposed and rigorously established. In [16] relaxation approximations are constructed by the introduction of special variables the so-called internal variables. In that setting applications to physical systems in elasticity and combustion theory are presented.…”