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

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 ε 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 dA(∂S) + o(√ε). The surface measure dA 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.