Energy-based fracture evolution

Over the past 15 years, there have been some significant successes in the mathematical analysis of fracture evolution, most notably proofs of existence for quasi-static (globally minimizing) Griffith evolutions. But for more physically reasonable settings, including even true quasi-static Griffith evolutions, existence is still open. This also applies to dynamic models, in which inertia is not neglected, which are particularly tricky to analyze, and even to formulate. I will describe the basic ideas and some of the early successes, as well as some of the reasons further progress is so difficult. I will then outline some recent results, and why these ideas have the potential to lead to substantial progress.