In my Dissertation, I consider the system of thermoelasticity endowed with poly
convex energy. I will present the equations in their mathematical and physical con
text, and I will explain the relevant research in the area and the contributions of my
work. First, I embed the equations of polyconvex thermoviscoelasticity into an aug
mented, symmetrizable, hyperbolic system which possesses a convex entropy. Using
the relative entropy method in the extended variables, I show convergence from ther
moviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth
solutions of the system of adiabatic thermoelasticity as both parameters tend to zero
and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity
in the zeroviscosity limit. In addition, I establish a weakstrong uniqueness result
for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.
Then, I prove a measurevalued versus strong uniqueness result for adiabatic poly
convex thermoelasticity in a suitable class of measurevalued solutions, de ned by
means of generalized Young measures that describe both oscillatory and concentra
tion e ects. Instead of working directly with the extended variables, I will look at
the parent system in the original variables utilizing the weak stability properties of
certain transportstretching identities, which allow to carry out the calculations by
placing minimal regularity assumptions in the energy framework. Next, I construct a
variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity.
I establish existence of minimizers which converge to a measurevalued solution that
dissipates the total energy. Also, I prove that the scheme converges when the limit
ing solution is smooth. Finally, for completeness and for the reader's convenience, I present the wellestablished theory for local existence of classical solutions and how
it applies to the equations at hand.
Date of Award  Nov 25 2020 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Athanasios Tzavaras (Supervisor) 

 conservation laws
 Thermoelasticity
 polyconvex
 partial differential equations