The equations of polyconvex thermoelasticity

Abstract

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.