A Relative Entropy Approach to Convergence of a Low Order Approximation to a Nonlinear Elasticity Model with Viscosity and Capillarity

Abstract

In this paper we study the dynamics of an elastic bar undergoing phase transitions. It is modeled by two regularizations of the equations of nonlinear elastodynamics with a nonconvex energy. We estimate the difference between solutions to the two regularizations if in one of them a coupling parameter is sent to infinity. This estimate is based on an adaptation of the relative entropy framework using the regularizing terms in order to compensate for the nonconvexity of the energy density. Introduction.This paper is concerned with two models describing longitudinal or shearing motions of an elastic bar undergoing phase transitions between a low strain and a high strain phase. The models are based on the (isothermal) equations of elastodynamics which, in the multiphase case, form a system of hyperbolic/elliptic conservation laws. It is well known that for such systems standard entropy conditions are insufficient to guarantee uniqueness of weak solutions. There are two approaches to overcoming this obstacle: One is to impose so-called kinetic relations at discontinuities, e.g., [1,21]. The other approach is to require solutions to be limits of solutions of regularized equations. We are interested in the second approach and study two such regularizations. One of these systems is well accepted in the literature, while the other offers computational advantages. We estimate the difference between solutions of the two regularized models. In particular, we will see that the regularizations compensate for the nonconvex energy density in that they make the models well-posed (which is well known and can be seen as the main reason for their introduction) and in that they allow us to use the relative entropy framework to derive estimates for the difference between solutions. To be more precise, let us introduce the two models. We consider the following third order model including viscous and capillary effects:

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

“…Here we provide some examples on how -in a physically meaningful and multi-dimensional situation -the higher-order (second gradient) regularization mechanism compensates for the non-convexity of the energy in such a way that the relative entropy technique still provides stability estimates. This extends results from [22], valid in a one-dimensional Lagrangian setting. To highlight the use of a stability theory for (1.12) derived via a modified relative entropy approach, we prove two results:…”

“…It should be compared to the well known relative entropy formulas initiated in the works Dafermos [11,12], DiPerna [15] and analogs that have been successfully used in many contexts (e.g. [27,30,20,8,28,22]). It has however a different origin from all these calculations: while the latter are based on the thermodynamical structure induced by the Clausius-Duhem inequality, the formula (1.7) is based on the abstract Hamiltonian flow structure in (1.1).…”

“…It has motivated efficient numerical schemes for the numerical treatment of diffuse interface systems for the description of phase transitions (see the discussion in the beginning of Section 5). The model convergence from (1.14) to (1.12) as α → ∞ was investigated in simple cases by [10] (using Fourier methods) and [22] (in a 1-d setting in Lagrangean coordinates). Here, we exploit the fact that the system (1.14) fits into the functional framework of (1.1), (1.5) and is thus equipped with a relative energy identity (see Section 2.6).…”

Stability properties of the Euler–Korteweg system with nonmonotone pressures

“…In [12,13] this was done for two strong solutions by estimating ∂ t e(ρ|ρ). However, here we only assume that ρ is a weak solution, so ∂ t e(ρ|ρ) is not well defined.…”

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

“…where (α 1 , α 2 ) is the interval of the decreasing pressures, i.e., p (s) < 0, see . We refer to [8,9,13,19,21] for first rigorous results on the Korteweg limit. It is possible to show that (2.8) formally converges to (2.1).…”