2024/07/09 by Michele Dolce, Dolce, Michele, Carl Johan Peter Johansson +3 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #35Q35 #35Q49 #76F25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2407.06884
openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the evolution of a passive scalar subject to molecular diffusion and advected by an incompressible velocity field on a 2D bounded domain. The velocity field is u = ∇^⊥ H, where H is an autonomous Hamiltonian whose level sets are Jordan curves foliating the domain. We focus on the high Péclet number regime (Pe := ν-1 ≫ 1), where two distinct processes unfold on well separated time-scales: streamline averaging and standard diffusion. For a specific class of Hamiltonians with one non-degenerate elliptic point (including perturbed radial flows), we prove exponential convergence of the solution to its streamline average on a subdiffusive time-scale Tν≪ ν-1 ,up to a small correction related to the shape of the streamlines. The time-scale Tν is determined by the behavior of the period function around the elliptic point. To establish this result, we introduce a model problem arising naturally from the difference between the solution and its streamline average. We use pseudospectral estimates to infer decay in the model problem, and, in fact, this analysis extends to a broader class of Hamiltonian flows. Finally, we perform an asymptotic expansion of the full solution, revealing that the leading terms consist of the streamline average and the solution of the model problem.