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 zero-viscosity limit. In addition, I establish a weak-strong uniqueness result
for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.
Then, I prove a measure-valued versus strong uniqueness result for adiabatic poly-
convex thermoelasticity in a suitable class of measure-valued 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 transport-stretching 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 measure-valued 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 well-established theory for local existence of classical solutions and how
it applies to the equations at hand.
| Identifer | oai:union.ndltd.org:kaust.edu.sa/oai:repository.kaust.edu.sa:10754/666127 |
| Date | 25 November 2020 |
| Creators | Galanopoulou, Myrto Maria |
| Contributors | Tzavaras, Athanasios, Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, Hoteit, Ibrahim, Markowich, Peter A., Christoforou, Cleopatra, Dafermos Constantine, M. |
| Source Sets | King Abdullah University of Science and Technology |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
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