The focusing problem for the porous medium equation: Experiment, simulation and analysis

The focusing problem involves the study of flows which are initially supported outside a compact empty region, with the aim of describing how the flows eventually fill the empty region. The work we describe has its origin in experimental studies of convergent gravity flows and in the global regularity theory of Caffarelli, Vázquez and Wolanski for the equivalent porous medium flow. We give a comprehensive survey of the current state of the experiments, numerical simulations and the analytic studies of this problem. The numerical studies involve innovative renormalization techniques and supply raw material for the analysis.