We consider a doubly nonlocal nonlinear parabolic equation which describes phase segregation of a binary system subject to weak-to-weak interactions [Gal, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018)]. The proposed model reduces to the classical Cahn–Hilliard equation under certain conditions. We establish well-posedness results (based on regular and nonregular mild solutions) along with regularity and long-time results in terms of finite-dimensional attractors. Then we also establish the convergence of (certain) mild solutions to single steady states as time goes to infinity. These results are also supplemented by a handful of (two-dimensional) numerical experiments displaying phase-segregation phenomena with interesting interface morphologies, depending on various choices of the interaction kernels (i.e., Gaussian, logarithmic, Riesz and bimodal potentials). We develop a stable numerical scheme which is able to control the computations under the effect of the double nonlinear convolutions.
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Keywords: Nonlocal Cahn–Hilliard, phase segregation, anomalous transport, doubly nonlocal equation, finite-dimensional attractor, numerical simulation
@article{AIHPC_2022__39_5_1179_0,
author = {Gal, Ciprian G. and Shomberg, Joseph L.},
title = {Cahn{\textendash}Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions},
journal = {Annales de l'Institut Henri Poincar\'e. C, Analyse non lin\'eaire},
pages = {1179--1234},
year = {2022},
volume = {39},
number = {5},
doi = {10.4171/aihpc/29},
language = {en},
url = {https://www.numdam.org/articles/10.4171/aihpc/29/}
}
TY - JOUR AU - Gal, Ciprian G. AU - Shomberg, Joseph L. TI - Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions JO - Annales de l'Institut Henri Poincaré. C, Analyse non linéaire PY - 2022 SP - 1179 EP - 1234 VL - 39 IS - 5 UR - https://www.numdam.org/articles/10.4171/aihpc/29/ DO - 10.4171/aihpc/29 LA - en ID - AIHPC_2022__39_5_1179_0 ER -
%0 Journal Article %A Gal, Ciprian G. %A Shomberg, Joseph L. %T Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions %J Annales de l'Institut Henri Poincaré. C, Analyse non linéaire %D 2022 %P 1179-1234 %V 39 %N 5 %U https://www.numdam.org/articles/10.4171/aihpc/29/ %R 10.4171/aihpc/29 %G en %F AIHPC_2022__39_5_1179_0
Gal, Ciprian G.; Shomberg, Joseph L. Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions. Annales de l'Institut Henri Poincaré. C, Analyse non linéaire, Tome 39 (2022) no. 5, pp. 1179-1234. doi: 10.4171/aihpc/29
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