Kolloquium GRK 2339, Prof. Robert Lazarzik

Energy-variational structure in evolution

Abstract: In this talk, we explore the energy-variational
framework for general evolution equations and their approximation
via a novel minimizing movement scheme. We present several recent
results that build on this structure, including existence theorems
for large classes of systems, selection principles, and
convergence results for numerical approximations.

Through a range of examples—including fluid dynamics, interface
evolution, and elasticity—we demonstrate how various
measure-valued solvability concepts from the literature can be
significantly simplified by exploiting the energy-variational
structure of the underlying partial differential equations. In
particular, we show that in several cases, a measure-valued
solution—typically represented by a high-dimensional Young
measure—can be equivalently reformulated using an
energy-variational solution involving only a scalar auxiliary
variable.

These examples reveal that such a structural property is not
incidental but reflects a recurring feature across many types of
evolution equations.