Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results

Abstract

AbstractWe consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of $$N\ge 2$$ N ≥ 2 components occupying a bounded region $$\Omega \subset \mathbb {R}^{d},$$ Ω ⊂ R d , $$d\le 3$$ d ≤ 3 . Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if $$d\le 2$$ d ≤ 2 , we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if $$d=3$$ d = 3 , we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If $$d=3$$ d = 3 , this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.