We consider the sharp interface limit of a coupled Stokes/Cahn–Hilliard system in a two-dimensional, bounded and smooth domain, i.e., we consider the limiting behavior of solutions when a parameter ε>0 corresponding to the thickness of the diffuse interface tends to zero. We show that for sufficiently short times the solutions to the Stokes/Cahn–Hilliard system converge to solutions of a sharp interface model, where the evolution of the interface is governed by a Mullins–Sekerka system with an additional convection term coupled to a two–phase stationary Stokes system with the Young–Laplace law for the jump of an extra contribution to the stress tensor, representing capillary stresses. We prove the convergence result by estimating the difference between the exact and an approximate solutions. To this end we make use of modifications of spectral estimates shown by X. Chen for the linearized Cahn–Hilliard operator. The treatment of the coupling terms requires careful estimates, the use of the refinements of the latter spectral estimate and a suitable structure of the approximate solutions, which will be constructed in the second part of this contribution.
Helmut Abels; Andreas Marquardt. Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result. Interfaces and free boundaries, Tome 23 (2021) no. 3, pp. 353-402. doi: 10.4171/ifb/457
@article{10_4171_ifb_457,
author = {Helmut Abels and Andreas Marquardt},
title = {Sharp interface limit of a {Stokes/Cahn{\textendash}Hilliard} system. {Part} {I:} {Convergence} result},
journal = {Interfaces and free boundaries},
pages = {353--402},
year = {2021},
volume = {23},
number = {3},
doi = {10.4171/ifb/457},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/457/}
}
TY - JOUR
AU - Helmut Abels
AU - Andreas Marquardt
TI - Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result
JO - Interfaces and free boundaries
PY - 2021
SP - 353
EP - 402
VL - 23
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/457/
DO - 10.4171/ifb/457
ID - 10_4171_ifb_457
ER -
%0 Journal Article
%A Helmut Abels
%A Andreas Marquardt
%T Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result
%J Interfaces and free boundaries
%D 2021
%P 353-402
%V 23
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/457/
%R 10.4171/ifb/457
%F 10_4171_ifb_457
