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    The equations of polyconvex thermoelasticity

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    Type
    Dissertation
    Authors
    Galanopoulou, Myrto Maria cc
    Advisors
    Tzavaras, Athanasios cc
    Committee members
    Hoteit, Ibrahim cc
    Markowich, Peter A. cc
    Christoforou, Cleopatra
    Dafermos Constantine, M.
    Program
    Applied Mathematics and Computational Science
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2020-11-25
    Permanent link to this record
    http://hdl.handle.net/10754/666127
    
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    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.
    Citation
    Galanopoulou, M. M. (2020). The equations of polyconvex thermoelasticity. KAUST Research Repository. https://doi.org/10.25781/KAUST-34G42
    DOI
    10.25781/KAUST-34G42
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-34G42
    Scopus Count
    Collections
    Applied Mathematics and Computational Science Program; Dissertations; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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