Johannes Zimmer: Moving phase boundaries: from microscopic models to continuum theory


One prominent example of microstructures occurs in so-called martensitic materials. These are phase transitions in metals where at low temperatures several stables configurations (“variants”) can coexist. To understand the time evolution of such materials, we need to understand the evolution of the underlying microstructure. Our present understanding of this phenomenon is quite limited, and we will focus of the simplest possible case of a single moving interface separating two stable states. As will be discussed in the lectures, the straightforward application of the elastodynamic setting (PDEs) leads to ill-posed equations. Therefore, we start with a microscopic (“atomistic”) model, namely a nonlocal Hamiltonian equation of motion. We discuss the rigorous existence theory of travelling interfaces under suitable assumptions. In a second part, a theory going back to Abeyaratne and Knowles and others will be described which allows us to single out special solutions of the equations of elastodynamics, thus rendering the equations of elastodynamics in this setting well-posed by providing a selection criterion. We will describe the current understanding how this selection criterion can be related to the microscopic theory described above, and sketch related open problems.