Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions

Abstract

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.