2021/02/16 by Nathanael Schillling, Schillling, Nathanael, Daniel Karrasch +3
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #35B25 #58J32 #58J35 #60G07 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and statistical mechanics #math-ph #math.AP #math.DG #math.MP #math.PR #msc:35B25 #msc:58J32 #msc:58J35 #msc:60G07
paper · pdf · doi:10.48550/arxiv.2102.08311
21 pages, 1 figure, submitted
openalex publication_date 2021/02/16 · arxiv created 2021/03/19 · arxiv updated 2021/03/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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 \mathds1S, is √(ε/π) dA(∂ 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.