Heat-content and diffusive leakage from material sets in the low-diffusivity limit ^*

Abstract

We generalize leading-order asymptotics of a form of the heat content of a submanifold (van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity ɛ goes to zero, the diffusive transport out of a material set S under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function 1 S , is ε / π d A ¯ ( ∂ S ) + o ( ε ) . The surface measure d A ¯ is that of the so-called geometry of mixing, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.

Detecting the birth and death of finite‐time coherent sets

“…Moreover, in addition to the stochastic trajectory and transfer operator interpretations in [22,23], we also provide a differential-geometric perspective. Other work arising from the dynamic Laplacian includes [38,39,54], where the emphasis is on the time-averaged processes generated by the dynamic Laplacian in the initial time slice on 𝑀.…”