  Finite-time coherent sets (FTCSs) are distinguished regions of phase space
that resist mixing with the surrounding space for some finite period of time;
physical manifestations include eddies and vortices in the ocean and
atmosphere, respectively. The boundaries of finite-time coherent sets are
examples of Lagrangian coherent structures (LCSs). The selection of the time
duration over which FTCS and LCS computations are made in practice is crucial
to their success. If this time is longer than the lifetime of coherence of
individual objects then existing methods will fail to detect the shorter-lived
coherence. It is of clear practical interest to determine the full lifetime of
coherent objects, but in complicated practical situations, for example a field
of ocean eddies with varying lifetimes, this is impossible with existing
approaches. Moreover, determining the timing of emergence and destruction of
coherent sets is of significant scientific interest. In this work we introduce
new constructions to address these issues. The key components are an inflated
dynamic Laplace operator and the concept of semi-material FTCSs. We make strong
mathematical connections between the inflated dynamic Laplacian and the
standard dynamic Laplacian [Froyland 2015], showing that the latter arises as a
limit of the former. The spectrum and eigenfunctions of the inflated dynamic
Laplacian directly provide information on the number, lifetimes, and evolution
of coherent~sets.
