The area of partial differential equations that concerns the mathematical theory of the equations
of thermomechanics has been a testing ground of several analytical ideas in the theory of nonlinear
partial differential equations and in turn a domain of development for numerical methods. The
advent of modern material science presents various new challenges to this area. The goal of this
minicourse is to present some of the analytical ideas and open problems in the area of thermomechanics,
with a goal to identify important elements of the structure of the equations.
Program wykˆadu:
- Introduction to the balance laws of continuum thermomechanics, transformations from Lagrangian
to Eulerian coordinates, the restrictions imposed to constitutive theories by the Clausius-
Duhem inequality and by the principle of frame indifference.
- Entropy pairs for systems of conservation laws, relative entropy and weak-strong uniqueness.
The equations of polyconvex elasticity, null-Lagrangians, the transport-stretching kinematic
identities, embedding polyconvex elastodynamics to a symmetric hyperbolic systems. Variational
approximation of the equations of elasticity, Applications to the existence of measure-valued
solutions for elasticity and to uniqueness of classical solutions in the class of entropic-mv solutions.
- Gradient flows in Wasserstein and the derivation of Fokker-Planck equations via the
Jordan-Kinderlehrer-Otto scheme, Gradient flows as large friction limit of Hamiltonian flows. Relative
entropy for the Hamiltonian flow. Applications to weak-strong uniqueness for the Euler-
Korteweg theory, relaxation limits from Euler with friction to porous media, and from Euler-
Korteweg to the Cahn-Hilliard equation.