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 sub-manifold (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 epsilon 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 root epsilon/pi d (A) over bar(partial derivative S) + o(root epsilon). 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.