A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space
Abstract
To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z ≥ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall—a "piston"—that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) ∼ ct^β with β ∈ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When β goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when β = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.
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BibTeX
@article{treton2026piston,
title = {{A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space}},
author = {Samuel Tréton and Mingmin Zhang},
journal = {SIAM Journal on Applied Mathematics},
year = {2026},
month = sep,
doi = {10.1137/25m1757435},
eprint = {2505.03304},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05054852v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space
AU - Samuel Tréton
AU - Mingmin Zhang
PY - 2026
DA - 2026/09/16
JO - SIAM Journal on Applied Mathematics
DO - 10.1137/25m1757435
UR - https://hal.science/hal-05054852v2
AB - To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z ≥ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall—a "piston"—that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) ∼ ct^β with β ∈ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When β goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when β = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.
SN - 0036-1399
SN - 1095-712X
ER -