Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges

The purpose of this article is to introduce the reader to phenomena on time-varying spatial domains and to highlight the differences from their counterpart on time-fixed domains. We begin by discussing the origin of this class of problems in various physical systems and applications, and then provide a general formulation from both Lagrangian and Eulerian viewpoints with the goal of identifying a set of basic principles necessary for understanding new effects on time-dependent domains. The distinctive features of the dynamics are illustrated with the help of two representative examples discussed in detail: (1) propagation of longitudinal waves in a stretching rod, and (2) Eckhaus instability of a stretching spatially periodic pattern. In view of the evolving character of the subject, we conclude with a number of open questions.