Maximum flows and minimum cuts in the plane

A continuous maximum flow problem finds the largest t such that div v = t F (x, y) is possible with a capacity constraint ‖(v1, v2)‖ ≤ c(x, y). The dual problem finds a minimum cut ∂S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph.