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We generalize leading-order asymptotics of a form of the  heat content of a submanifold  (van den Berg &amp; 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 &amp; Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.

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

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