Roughening it — evolving irregular domains and transport theorems

Abstract

Well-known examples of transport theorems include the Leibniz integral rule and a result established by Reynolds for three-dimensional regions that convect with the motion of a continuum. Here, we prove a generalized transport theorem by relaxing the regularity assumptions on the domain of integration to include domains that may, among other things, develop holes, split into pieces, or whose fractal dimension may evolve with time. Our results are of potential value in continuum physics and the calculus of variations.