Communications in Information and Systems

Volume 13 (2013)

Number 4

Special Issue in Honor of Marshall Slemrod: Part 4 of 4

Stress relaxation models with polyconvex entropy in Lagrangean and Eulerian coordinates

Pages: 487 – 515



Athanasios E. Tzavaras (Department of Applied Mathematics, University of Crete, Heraklion, Greece; Institute for Applied and Computational Mathematics, FORTH, Heraklion, Greece)


The embedding of the equations of polyconvex elastodynamics to an augmented symmetric hyperbolic system provides in conjunction with the relative entropy method a robust stability framework for approximate solutions. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using the relative entropy we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime. As an application, we show that models of pressure relaxation for real gases in Eulerian coordinates fit into the proposed framework.


polyconvex elasticity, relaxation limits, pressure relaxation in gases

2010 Mathematics Subject Classification

35L65, 35L75, 74B20, 82C40

Full Text (PDF format)