Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

Cleopatra Christoforou; Myrto Galanopoulou; Athanasios E. Tzavaras

doi:10.3934/dcds.2019269

Article Contents

2019, Volume 39, Issue 11: 6175-6206. Doi: 10.3934/dcds.2019269

This issue Previous Article Next Article Measure dynamics with Probability Vector Fields and sources

Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

1.
Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus
2.
Computer, Electrical, Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

^* Corresponding author: Christoforou was partially supported by the Internal grant SBLawsMechGeom #21036 from University of Cyprus
^* Corresponding author: Christoforou was partially supported by the Internal grant SBLawsMechGeom #21036 from University of Cyprus

Received: August 2018

Revised: November 2018

Published: August 2019

This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No 642768

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Abstract

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. Exploiting the weak-stability properties of the transport and stretching identities, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.

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Mathematics Subject Classification: Primary: 35Q74; Secondary: 35A02, 35L65, 74B20, 74D10, 80A17, 74H25.

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References

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