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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.
@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'I.H.P. Analyse non lin\'eaire},
pages = {1179--1234},
volume = {39},
number = {5},
year = {2022},
doi = {10.4171/aihpc/29},
language = {en},
url = {http://geodesic.mathdoc.fr/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'I.H.P. Analyse non linéaire PY - 2022 SP - 1179 EP - 1234 VL - 39 IS - 5 UR - http://geodesic.mathdoc.fr/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'I.H.P. Analyse non linéaire %D 2022 %P 1179-1234 %V 39 %N 5 %U http://geodesic.mathdoc.fr/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'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 5, pp. 1179-1234. doi: 10.4171/aihpc/29
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