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Long-time behavior of
the Stokes-transport system in a channel
Anne-Laure Dalibard, Julien Guillod and Antoine Leblond |
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Vol. 18 (2025), No. 8, 1955–2032
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Abstract |
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We consider here a two-dimensional incompressible fluid in a periodic channel, whose density is advected by pure transport, and whose velocity is given by the Stokes equation with gravity source term. Dirichlet boundary conditions are taken for the velocity field on the bottom and top of the channel and periodic conditions in the horizontal variable. We prove that the affine stratified density profile is stable under small perturbations in Sobolev spaces and prove convergence of the density to another limiting stratified density profile for large time with an explicit algebraic decay rate. Moreover, we are able to precisely identify the limiting profile as the decreasing vertical rearrangement of the initial density. Finally, we show that boundary layers are formed for large times in the vicinity of the upper and lower boundaries. These boundary layers, which had not been identified in previous works, are given by a self-similar ansatz and driven by a linear mechanism. This allows us to precisely characterize the long-time behavior beyond the constant limiting profile and reach more optimal decay rates. |
Keywords
Stokes-transport, channel, stability, affine profile,
decreasing vertical rearrangement, long-time behavior,
boundary layers, Dirichlet boundary conditions, optimal
decay rate
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Mathematical Subject Classification
Primary: 35B35, 35B40, 35Q49, 76D07, 76D10
Secondary: 35D35, 35M13
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Milestones
Received: 24 November 2023
Accepted: 23 September 2024
Published: 25 July 2025
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Authors |
| Anne-Laure Dalibard | |
| Laboratoire Jacques-Louis Lions, UMR
CNRS 7598 Sorbonne Université Université Paris Cité, Inria Paris France |
Département de Mathématiques et
applications École Normale Supérieure Université PSL Paris France |
| Julien Guillod | |
| Laboratoire Jacques-Louis Lions, UMR
CNRS 7598 Sorbonne Université Université Paris Cité, Inria Paris France |
Département de Mathématiques et
applications École Normale Supérieure Université PSL Paris France |
| Antoine Leblond | |
| Laboratoire Jacques-Louis Lions, UMR
CNRS 7598 Sorbonne Université Université Paris Cité, Inria Paris France |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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