Séminaire EDP-Analyse ICJ

Long-time behavior of the Stokes-transport system in a channel.

by Antoine Leblond (Max Planck - Hambourg)

Fokko (ICJ)




We consider here a 2d 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 show 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 enlighten the optimal decay rate.

This is a joint work with Anne-Laure Dalibard and Julien Guillod (Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France), part of my PhD thesis.